Linear Equations and Inequalities

Linear equations and inequalities form the foundation of algebra and are widely used in various real-world applications. A linear equation is an equation that makes a straight line when graphed. It contains variables raised to the power of 1, and its general form is ax+b=c. A linear inequality is similar to a linear equation but involves inequalities such as >, <, , or instead of an equal sign.

This topic will cover the theory behind linear equations and inequalities, explain how to solve them, and provide 100 examples to solidify the concepts.


1. What is a Linear Equation?

A linear equation is any equation that can be written in the form ax+b=c, where x is the variable, and a, b, and c are constants. The highest power of the variable is 1, which means the equation is linear.

For example:

  • 3x+5=11 is a linear equation.
  • x4=8 is another linear equation.

Steps to Solve Linear Equations:

  1. Simplify: Simplify both sides of the equation if necessary by combining like terms.
  2. Isolate the Variable: Use inverse operations (addition, subtraction, multiplication, or division) to get the variable on one side of the equation.
  3. Solve for the Variable: Perform the required operations to find the value of the variable.

Example: Solve 3x+5=11.

Solution:
3x+5=11
3x=115
3x=6
x=63
x=2


2. What is a Linear Inequality?

A linear inequality is similar to a linear equation but involves an inequality symbol (<, >, , or ) instead of an equal sign. The general form is ax+b<c, where x is the variable and a, b, and c are constants.

For example:

  • 2x+3<7 is a linear inequality.
  • 5x19 is another linear inequality.

Steps to Solve Linear Inequalities:

  1. Simplify: Simplify both sides of the inequality.
  2. Isolate the Variable: Use inverse operations to move the variable to one side.
  3. Solve for the Variable: Perform the necessary operations.
  4. Flip the Sign: When multiplying or dividing both sides of an inequality by a negative number, remember to flip the inequality sign.

Example: Solve 2x+3<7.

Solution:
2x+3<7
2x<73
2x<4
x<42
x<2


3. Solving Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. The solution is the set of values that satisfy all the equations simultaneously.

Methods to Solve Systems of Equations:

  1. Substitution: Solve one equation for one variable and substitute it into the other equation.
  2. Elimination: Add or subtract equations to eliminate one variable, then solve for the remaining variable.
  3. Graphing: Graph both equations and find the point where they intersect.

Example: Solve the system:
x+y=5
2xy=3

Solution using substitution:
From the first equation:
y=5x

Substitute this into the second equation:
2x(5x)=3
2x5+x=3
3x5=3
3x=3+5
3x=8
x=83

Now, substitute x=83 into y=5x:
y=583
y=15383
y=73

Thus, the solution is x=83 and y=73.


4. Graphing Linear Inequalities

Graphing linear inequalities involves shading a region on a graph that represents all the possible solutions. To graph a linear inequality:

  1. First, graph the boundary line (the equation obtained by replacing the inequality sign with an equal sign).
  2. Determine whether to use a solid line (for or ) or a dashed line (for < or >).
  3. Test a point that is not on the boundary line to decide which side of the line to shade.

Example: Graph the inequality 3xy4.

Solution:

  1. Graph the boundary line 3xy=4.
  2. Since the inequality symbol is , use a solid line.
  3. Test the point (0,0):
    3(0)(0)=0, which is not greater than or equal to 4.
    So, shade the region above the line.

5. 100 Examples of Linear Equations and Inequalities


Examples 1 to 10


Example 1: Solve the equation 4x+2=10.

Solution:
4x+2=10
4x=102
4x=8
x=84
x=2


Example 2: Solve the inequality 3x5<10.

Solution:
3x5<10
3x<10+5
3x<15
x<153
x<5


Example 3: Solve the system of equations:
2x+y=7
xy=1

Solution using substitution:
From the second equation:
y=x1

Substitute into the first equation:
2x+(x1)=7
2x+x1=7
3x1=7
3x=7+1
3x=8
x=83

Now, substitute x=83 into y=x1:
y=831
y=8333
y=53

Thus, the solution is x=83 and y=53.


Example 4: Solve the inequality 5x+318.

Solution:
5x+318
5x183
5x15
x155
x3


Example 5: Solve 2x+5=3x4.

Solution:
2x+5=3x4
2x3x=45
x=9
x=9


Example 6: Solve the inequality 6x2>10.

Solution:
6x2>10
6x>10+2
6x>12
x>126
x>2


Example 7: Solve the system of equations:
x+2y=8
2xy=3

Solution using elimination:
From the second equation, multiply by 2: 2(x+2y=8)2x+4y=16
Now, add the two equations: 2xy=3
2x+4y=16
By adding these, we get: (2x+4y)+(2xy)=16+3
4x+3y=19.

Solve for y by substitution.


Example 8: Solve the inequality 7x325.

Solution:
7x325
7x25+3
7x28
x287
x4


Example 9: Solve 3x+4=5x2.

Solution:
3x+4=5x2
3x5x=24
2x=6
x=62
x=3


Example 10: Solve the inequality 4x+517.

Solution:
4x+517
4x175
4x12
x124
x3


Examples 11 to 20


Example 11: Solve the system of equations:
2x+y=6
3x2y=7

Solution using substitution:
From the first equation: y=62x.

Substitute into the second equation: 3x2(62x)=7
3x12+4x=7
7x12=7
7x=19
x=197.

Now substitute x=197 into y=62x: y=62(197)
y=427387
y=47.

Thus, x=197 and y=47.


Example 12: Solve the inequality 6x9<15.

Solution:
6x9<15
6x<15+9
6x<24
x<246
x<4


Example 13: Solve 5x7=2x+8.

Solution:
5x7=2x+8
5x2x=8+7
3x=15
x=153
x=5


Example 14: Solve the inequality 4x+723.

Solution:
4x+723
4x237
4x16
x164
x4


Example 15: Solve the system of equations:
xy=2
2x+y=7

Solution using substitution:
From the first equation: x=y+2.

Substitute into the second equation: 2(y+2)+y=7
2y+4+y=7
3y+4=7
3y=74
3y=3
y=33
y=1.

Substitute y=1 into x=y+2: x=1+2=3.

Thus, the solution is x=3 and y=1.


Example 16: Solve the inequality 2x+511.

Solution:
2x+511
2x115
2x6
x62
x3


Example 17: Solve 7x3=4x+6.

Solution:
7x3=4x+6
7x4x=6+3
3x=9
x=93
x=3


Example 18: Solve the inequality 5x+4<20.

Solution:
5x+4<20
5x<204
5x<16
x<165
x<3.2


Example 19: Solve the system of equations:
3x+y=11
xy=2

Solution using substitution:
From the second equation: x=y+2.

Substitute into the first equation: 3(y+2)+y=11
3y+6+y=11
4y+6=11
4y=116
4y=5
y=54.

Now, substitute y=54 into x=y+2: x=54+2
x=54+84
x=134.

Thus, x=134 and y=54.


Example 20: Solve the inequality 7x219.

Solution:
7x219
7x19+2
7x21
x217
x3


Examples 21 to 30


Example 21: Solve the system of equations:
2x+3y=12
xy=2

Solution using substitution:
From the second equation:
x=y+2.

Substitute into the first equation:
2(y+2)+3y=12
2y+4+3y=12
5y+4=12
5y=124
5y=8
y=85.

Now substitute y=85 into x=y+2:
x=85+2
x=85+105
x=185.

Thus, x=185 and y=85.


Example 22: Solve the inequality 5x3<12.

Solution:
5x3<12
5x<12+3
5x<15
x<155
x<3


Example 23: Solve 4x+6=2x+12.

Solution:
4x+6=2x+12
4x2x=126
2x=6
x=62
x=3


Example 24: Solve the inequality 3x+716.

Solution:
3x+716
3x167
3x9
x93
x3


Example 25: Solve the system of equations:
x+3y=9
2xy=7

Solution using substitution:
From the first equation:
x=93y.

Substitute into the second equation:
2(93y)y=7
186yy=7
187y=7
7y=187
7y=11
y=117.

Substitute y=117 into x=93y:
x=93(117)
x=637337
x=307.

Thus, x=307 and y=117.


Example 26: Solve the inequality 4x511.

Solution:
4x511
4x11+5
4x16
x164
x4


Example 27: Solve 6x7=4x+9.

Solution:
6x7=4x+9
6x4x=9+7
2x=16
x=162
x=8


Example 28: Solve the inequality 3x+2<14.

Solution:
3x+2<14
3x<142
3x<12
x<123
x<4


Example 29: Solve the system of equations:
2x+5y=15
3x4y=11

Solution using elimination:
Multiply the first equation by 3 and the second equation by 2:
3(2x+5y=15)6x+15y=45
2(3x4y=11)6x8y=22.

Subtract the second equation from the first:
(6x+15y)(6x8y)=4522
23y=23
y=2323
y=1.

Substitute y=1 into 2x+5y=15:
2x+5(1)=15
2x+5=15
2x=155
2x=10
x=102
x=5.

Thus, x=5 and y=1.


Example 30: Solve the inequality 8x321.

Solution:
8x321
8x21+3
8x24
x248
x3


Examples 31 to 40


Example 31: Solve the system of equations:
x+2y=7
2xy=4

Solution using substitution:
From the first equation:
x=72y.

Substitute into the second equation:
2(72y)y=4
144yy=4
145y=4
5y=144
5y=10
y=105
y=2.

Now, substitute y=2 into x=72y:
x=72(2)
x=74
x=3.

Thus, x=3 and y=2.


Example 32: Solve the inequality 5x+7<22.

Solution:
5x+7<22
5x<227
5x<15
x<155
x<3


Example 33: Solve the inequality 3x27.

Solution:
3x27
3x7+2
3x9
x93
x3


Example 34: Solve the system of equations:
4x+y=12
2xy=4

Solution using elimination:
Add both equations:
(4x+y)+(2xy)=12+4
6x=16
x=166=83.

Substitute x=83 into 4x+y=12:
4(83)+y=12
323+y=12
y=12323
y=363323
y=43.

Thus, x=83 and y=43.


Example 35: Solve the inequality 7x+4<25.

Solution:
7x+4<25
7x<254
7x<21
x<217
x<3


Example 36: Solve the system of equations:
5x+2y=20
3x4y=10

Solution using substitution:
From the first equation:
5x+2y=20x=202y5.

Substitute into the second equation:
3(202y5)4y=10
606y54y=10
Multiply by 5 to eliminate the fraction:
606y20y=50
26y=5060
26y=10
y=1026=513.

Substitute y=513 into x=202y5:
x=202(513)5
x=2010135
x=2601310135
x=25013×5
x=25065
x=5013.

Thus, x=5013 and y=513.


Example 37: Solve the inequality 4x79.

Solution:
4x79
4x9+7
4x16
x164
x4


Example 38: Solve the system of equations:
x+3y=6
2xy=4

Solution using substitution:
From the first equation:
x=63y.

Substitute into the second equation:
2(63y)y=4
126yy=4
127y=4
7y=124
7y=8
y=87.

Substitute y=87 into x=63y:
x=63(87)
x=427247
x=187.

Thus, x=187 and y=87.


Example 39: Solve the inequality 2x+5<18.

Solution:
2x+5<18
2x<185
2x<13
x<132
x<6.5


Example 40: Solve the system of equations:
3x+4y=24
2x3y=6

Solution using elimination:
Multiply the first equation by 3 and the second equation by 4:
3(3x+4y=24)9x+12y=72
4(2x3y=6)8x12y=24.

Add both equations:
(9x+12y)+(8x12y)=72+24
17x=96
x=9617
x=9617.

Now substitute x=9617 into 3x+4y=24:
3(9617)+4y=24
28817+4y=24
4y=2428817
4y=4081728817
4y=12017
y=12017×4
y=3017.

Thus, x=9617 and y=3017.


Examples 41 to 50


Example 41: Solve the system of equations:
x2y=4
3x+y=11

Solution using substitution:
From the first equation:
x=4+2y.

Substitute into the second equation:
3(4+2y)+y=11
12+6y+y=11
12+7y=11
7y=1112
7y=1
y=17.

Substitute y=17 into x=4+2y:
x=4+2(17)
x=427
x=28727
x=267.

Thus, x=267 and y=17.


Example 42: Solve the inequality 5x416.

Solution:
5x416
5x16+4
5x20
x205
x4


Example 43: Solve the system of equations:
4x+3y=21
xy=2

Solution using elimination:
Multiply the second equation by 3:
3(xy=2)3x3y=6.

Now add the two equations:
4x+3y+3x3y=21+6
7x=27
x=277.

Substitute x=277 into xy=2:
277y=2
y=2772
y=277147
y=137.

Thus, x=277 and y=137.


Example 44: Solve the inequality 3x+7<19.

Solution:
3x+7<19
3x<197
3x<12
x<123
x<4


Example 45: Solve the system of equations:
2x+y=7
5x2y=11

Solution using substitution:
From the first equation:
y=72x.

Substitute into the second equation:
5x2(72x)=11
5x14+4x=11
9x14=11
9x=11+14
9x=25
x=259.

Substitute x=259 into y=72x:
y=72(259)
y=7509
y=639509
y=139.

Thus, x=259 and y=139.


Example 46: Solve the inequality 6x39.

Solution:
6x39
6x9+3
6x12
x126
x2


Example 47: Solve the system of equations:
3x+y=9
2xy=4

Solution using elimination:
Add both equations:
(3x+y)+(2xy)=9+4
5x=13
x=135.

Substitute x=135 into 3x+y=9:
3(135)+y=9
395+y=9
y=9395
y=455395
y=65.

Thus, x=135 and y=65.


Example 48: Solve the inequality 8x+4<28.

Solution:
8x+4<28
8x<284
8x<24
x<248
x<3


Example 49: Solve the system of equations:
x+4y=14
3x2y=12

Solution using substitution:
From the first equation:
x=144y.

Substitute into the second equation:
3(144y)2y=12
4212y2y=12
4214y=12
14y=1242
14y=30
y=3014=157.

Substitute y=157 into x=144y:
x=144(157)
x=987607
x=387.

Thus, x=387 and y=157.


Example 50: Solve the inequality 7x516.

Solution:
7x516
7x16+5
7x21
x217
x3


Examples 51 to 60


Example 51: Solve the system of equations:
2x+3y=13
xy=2

Solution using substitution:
From the second equation:
x=y+2.

Substitute into the first equation:
2(y+2)+3y=13
2y+4+3y=13
5y+4=13
5y=134
5y=9
y=95.

Substitute y=95 into x=y+2:
x=95+2
x=95+105
x=195.

Thus, x=195 and y=95.


Example 52: Solve the inequality 4x+618.

Solution:
4x+618
4x186
4x12
x124
x3


Example 53: Solve the system of equations:
5x+y=20
2xy=4

Solution using elimination:
Add both equations:
(5x+y)+(2xy)=20+4
7x=24
x=247.

Substitute x=247 into 5x+y=20:
5(247)+y=20
1207+y=20
y=201207
y=14071207
y=207.

Thus, x=247 and y=207.


Example 54: Solve the inequality 7x9<12.

Solution:
7x9<12
7x<12+9
7x<21
x<217
x<3


Example 55: Solve the system of equations:
3x+y=10
4xy=12

Solution using elimination:
Add both equations:
(3x+y)+(4xy)=10+12
7x=22
x=227.

Substitute x=227 into 3x+y=10:
3(227)+y=10
667+y=10
y=10667
y=707667
y=47.

Thus, x=227 and y=47.


Example 56: Solve the inequality 8x+428.

Solution:
8x+428
8x284
8x24
x248
x3


Example 57: Solve the system of equations:
6x+2y=18
4x3y=10

Solution using substitution:
From the first equation:
6x+2y=18x=182y6.

Substitute into the second equation:
4(182y6)3y=10
728y63y=10
Multiply by 6 to eliminate the fraction:
728y18y=60
7226y=60
26y=6072
26y=12
y=1226=613.

Substitute y=613 into x=182y6:
x=182(613)6
x=1812136
x=2341312136
x=22213×6
x=11139.

Thus, x=11139 and y=613.


Example 58: Solve the inequality 5x7<18.

Solution:
5x7<18
5x<18+7
5x<25
x<255
x<5


Example 59: Solve the system of equations:
x+3y=11
2xy=4

Solution using substitution:
From the first equation:
x=113y.

Substitute into the second equation:
2(113y)y=4
226yy=4
227y=4
7y=224
7y=18
y=187.

Substitute y=187 into x=113y:
x=113(187)
x=777547
x=237.

Thus, x=237 and y=187.


Example 60: Solve the inequality 4x+315.

Solution:
4x+315
4x153
4x12
x124
x3


Examples 61 to 70


Example 61: Solve the system of equations:
2x+5y=17
3x2y=4

Solution using substitution:
From the first equation:
x=175y2.

Substitute into the second equation:
3(175y2)2y=4
5115y22y=4
Multiply by 2 to eliminate the fraction:
5115y4y=8
5119y=8
19y=851
19y=43
y=4319=4319.

Substitute y=4319 into x=175y2:
x=175(4319)2
x=17215192
x=32319215192
x=10819×2
x=10838
x=5419.

Thus, x=5419 and y=4319.


Example 62: Solve the inequality 6x+826.

Solution:
6x+826
6x268
6x18
x186
x3


Example 63: Solve the system of equations:
4x+y=13
2xy=3

Solution using elimination:
Add both equations:
(4x+y)+(2xy)=13+3
6x=16
x=166=83.

Substitute x=83 into 4x+y=13:
4(83)+y=13
323+y=13
y=13323
y=393323
y=73.

Thus, x=83 and y=73.


Example 64: Solve the inequality 4x93.

Solution:
4x93
4x3+9
4x12
x124
x3


Example 65: Solve the system of equations:
5x+y=14
3x2y=7

Solution using substitution:
From the first equation:
y=145x.

Substitute into the second equation:
3x2(145x)=7
3x28+10x=7
13x28=7
13x=7+28
13x=35
x=3513.

Substitute x=3513 into y=145x:
y=145(3513)
y=1821317513
y=713.

Thus, x=3513 and y=713.


Example 66: Solve the inequality 5x6<9.

Solution:
5x6<9
5x<9+6
5x<15
x<155
x<3


Example 67: Solve the system of equations:
2x+3y=9
4xy=7

Solution using substitution:
From the second equation:
y=4x7.

Substitute into the first equation:
2x+3(4x7)=9
2x+12x21=9
14x21=9
14x=9+21
14x=30
x=3014=157.

Substitute x=157 into y=4x7:
y=4(157)7
y=6077
y=607497
y=117.

Thus, x=157 and y=117.


Example 68: Solve the inequality 3x+511.

Solution:
3x+511
3x115
3x6
x63
x2


Example 69: Solve the system of equations:
3x+2y=10
xy=2

Solution using substitution:
From the second equation:
y=x2.

Substitute into the first equation:
3x+2(x2)=10
3x+2x4=10
5x4=10
5x=10+4
5x=14
x=145.

Substitute x=145 into y=x2:
y=1452
y=145105
y=45.

Thus, x=145 and y=45.


Example 70: Solve the inequality 6x8<16.

Solution:
6x8<16
6x<16+8
6x<24
x<246
x<4


Examples 71 to 80


Example 71: Solve the system of equations:
5x+2y=14
4xy=5

Solution using substitution:
From the second equation:
y=4x5.

Substitute into the first equation:
5x+2(4x5)=14
5x+8x10=14
13x10=14
13x=14+10
13x=24
x=2413.

Substitute x=2413 into y=4x5:
y=4(2413)5
y=96135
y=96136513
y=3113.

Thus, x=2413 and y=3113.


Example 72: Solve the inequality 8x915.

Solution:
8x915
8x15+9
8x24
x248
x3


Example 73: Solve the system of equations:
3x+5y=19
2xy=6

Solution using elimination:
Multiply the second equation by 5:
5(2xy=6)10x5y=30.

Now add both equations:
(3x+5y)+(10x5y)=19+30
13x=49
x=4913.

Substitute x=4913 into 2xy=6:
2(4913)y=6
9813y=6
y=98136
y=98137813
y=2013.

Thus, x=4913 and y=2013.


Example 74: Solve the inequality 7x+4<23.

Solution:
7x+4<23
7x<234
7x<19
x<197
x<2.71


Example 75: Solve the system of equations:
2x+4y=12
3xy=5

Solution using substitution:
From the first equation:
x=124y2.

Substitute into the second equation:
3(124y2)y=5
3612y2y=5
Multiply by 2 to eliminate the fraction:
3612y2y=10
3614y=10
14y=1036
14y=26
y=2614=137.

Substitute y=137 into x=124y2:
x=124(137)2
x=125272
x=8475272
x=327×2
x=3214=167.

Thus, x=167 and y=137.


Example 76: Solve the inequality 5x87.

Solution:
5x87
5x7+8
5x15
x155
x3


Example 77: Solve the system of equations:
4x+y=11
2xy=3

Solution using elimination:
Add both equations:
(4x+y)+(2xy)=11+3
6x=14
x=146=73.

Substitute x=73 into 4x+y=11:
4(73)+y=11
283+y=11
y=11283
y=333283
y=53.

Thus, x=73 and y=53.


Example 78: Solve the inequality 4x+921.

Solution:
4x+921
4x219
4x12
x124
x3


Example 79: Solve the system of equations:
x+2y=5
3xy=8

Solution using substitution:
From the first equation:
x=52y.

Substitute into the second equation:
3(52y)y=8
156yy=8
157y=8
7y=158
7y=7
y=77
y=1.

Substitute y=1 into x=52y:
x=52(1)
x=52
x=3.

Thus, x=3 and y=1.


Example 80: Solve the inequality 6x4<14.

Solution:
6x4<14
6x<14+4
6x<18
x<186
x<3


Examples 81 to 90


Example 81: Solve the system of equations:
2x+3y=15
x4y=1

Solution using substitution:
From the second equation:
x=4y1.

Substitute into the first equation:
2(4y1)+3y=15
8y2+3y=15
11y2=15
11y=15+2
11y=17
y=1711.

Substitute y=1711 into x=4y1:
x=4(1711)1
x=68111
x=68111111
x=5711.

Thus, x=5711 and y=1711.


Example 82: Solve the inequality 3x+419.

Solution:
3x+419
3x194
3x15
x153
x5


Example 83: Solve the system of equations:
5x+2y=18
3xy=7

Solution using elimination:
Multiply the second equation by 2:
2(3xy=7)6x2y=14.

Now add both equations:
(5x+2y)+(6x2y)=18+14
11x=32
x=3211.

Substitute x=3211 into 3xy=7:
3(3211)y=7
9611y=7
y=96117
y=96117711
y=1911.

Thus, x=3211 and y=1911.


Example 84: Solve the inequality 5x3<12.

Solution:
5x3<12
5x<12+3
5x<15
x<155
x<3


Example 85: Solve the system of equations:
4x+y=19
2xy=1

Solution using elimination:
Add both equations:
(4x+y)+(2xy)=19+1
6x=20
x=206=103.

Substitute x=103 into 4x+y=19:
4(103)+y=19
403+y=19
y=19403
y=573403
y=173.

Thus, x=103 and y=173.


Example 86: Solve the inequality 6x711.

Solution:
6x711
6x11+7
6x18
x186
x3


Example 87: Solve the system of equations:
x+2y=8
2xy=3

Solution using substitution:
From the first equation:
x=82y.

Substitute into the second equation:
2(82y)y=3
164yy=3
165y=3
5y=163
5y=13
y=135.

Substitute y=135 into x=82y:
x=82(135)
x=8265
x=405265
x=145.

Thus, x=145 and y=135.


Example 88: Solve the inequality 4x+721.

Solution:
4x+721
4x217
4x14
x144
x3.5


Example 89: Solve the system of equations:
2x+y=10
5x3y=1

Solution using substitution:
From the first equation:
y=102x.

Substitute into the second equation:
5x3(102x)=1
5x30+6x=1
11x30=1
11x=1+30
11x=31
x=3111.

Substitute x=3111 into y=102x:
y=102(3111)
y=106211
y=110116211
y=4811.

Thus, x=3111 and y=4811.


Example 90: Solve the inequality 3x8<7.

Solution:
3x8<7
3x<7+8
3x<15
x<153
x<5


Linear Equations and Inequalities

Linear equations and inequalities form the foundation of algebra and are widely used in various real-world applications. A linear equation is an equation that makes a straight line when graphed. It contains variables raised to the power of 1, and its general form is ax+b=c. A linear inequality is similar to a linear equation but involves inequalities such as >, <, , or instead of an equal sign.

This topic will cover the theory behind linear equations and inequalities, explain how to solve them, and provide 100 examples to solidify the concepts.


1. What is a Linear Equation?

A linear equation is any equation that can be written in the form ax+b=c, where x is the variable, and a, b, and c are constants. The highest power of the variable is 1, which means the equation is linear.

For example:

  • 3x+5=11 is a linear equation.
  • x4=8 is another linear equation.

Steps to Solve Linear Equations:

  1. Simplify: Simplify both sides of the equation if necessary by combining like terms.
  2. Isolate the Variable: Use inverse operations (addition, subtraction, multiplication, or division) to get the variable on one side of the equation.
  3. Solve for the Variable: Perform the required operations to find the value of the variable.

Example: Solve 3x+5=11.

Solution:
3x+5=11
3x=115
3x=6
x=63
x=2


2. What is a Linear Inequality?

A linear inequality is similar to a linear equation but involves an inequality symbol (<, >, , or ) instead of an equal sign. The general form is ax+b<c, where x is the variable and a, b, and c are constants.

For example:

  • 2x+3<7 is a linear inequality.
  • 5x19 is another linear inequality.

Steps to Solve Linear Inequalities:

  1. Simplify: Simplify both sides of the inequality.
  2. Isolate the Variable: Use inverse operations to move the variable to one side.
  3. Solve for the Variable: Perform the necessary operations.
  4. Flip the Sign: When multiplying or dividing both sides of an inequality by a negative number, remember to flip the inequality sign.

Example: Solve 2x+3<7.

Solution:
2x+3<7
2x<73
2x<4
x<42
x<2


3. Solving Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. The solution is the set of values that satisfy all the equations simultaneously.

Methods to Solve Systems of Equations:

  1. Substitution: Solve one equation for one variable and substitute it into the other equation.
  2. Elimination: Add or subtract equations to eliminate one variable, then solve for the remaining variable.
  3. Graphing: Graph both equations and find the point where they intersect.

Example: Solve the system:
x+y=5
2xy=3

Solution using substitution:
From the first equation:
y=5x

Substitute this into the second equation:
2x(5x)=3
2x5+x=3
3x5=3
3x=3+5
3x=8
x=83

Now, substitute x=83 into y=5x:
y=583
y=15383
y=73

Thus, the solution is x=83 and y=73.


4. Graphing Linear Inequalities

Graphing linear inequalities involves shading a region on a graph that represents all the possible solutions. To graph a linear inequality:

  1. First, graph the boundary line (the equation obtained by replacing the inequality sign with an equal sign).
  2. Determine whether to use a solid line (for or ) or a dashed line (for < or >).
  3. Test a point that is not on the boundary line to decide which side of the line to shade.

Example: Graph the inequality 3xy4.

Solution:

  1. Graph the boundary line 3xy=4.
  2. Since the inequality symbol is , use a solid line.
  3. Test the point (0,0):
    3(0)(0)=0, which is not greater than or equal to 4.
    So, shade the region above the line.

5. 100 Examples of Linear Equations and Inequalities


Examples 1 to 10


Example 1: Solve the equation 4x+2=10.

Solution:
4x+2=10
4x=102
4x=8
x=84
x=2


Example 2: Solve the inequality 3x5<10.

Solution:
3x5<10
3x<10+5
3x<15
x<153
x<5


Examples 91 to 100


Example 91: Solve the system of equations:
3x+4y=20
2xy=7

Solution using substitution:
From the second equation:
y=2x7.

Substitute into the first equation:
3x+4(2x7)=20
3x+8x28=20
11x28=20
11x=20+28
11x=48
x=4811.

Substitute x=4811 into y=2x7:
y=2(4811)7
y=96117
y=96117711
y=1911.

Thus, x=4811 and y=1911.


Example 92: Solve the inequality 5x+924.

Solution:
5x+924
5x249
5x15
x155
x3


Example 93: Solve the system of equations:
4x+6y=22
3x2y=5

Solution using elimination:
Multiply the second equation by 3 and the first equation by 2:
3(3x2y=5)9x6y=15
2(4x+6y=22)8x+12y=44.

Add both equations:
9x6y+8x+12y=15+44
17x+6y=59
x=5917.

Substitute x=5917 into 4x+6y=22:
4(5917)+6y=22
23617+6y=22
6y=2223617
y=3741723617
y=13817.

Thus, x=5917 and y=13817.


Example 94: Solve the inequality 4x6<16.

Solution:
4x6<16
4x<16+6
4x<22
x<224
x<5.5


Example 95: Solve the system of equations:
3x+2y=17
5xy=14

Solution using substitution:
From the second equation:
y=5x14.

Substitute into the first equation:
3x+2(5x14)=17
3x+10x28=17
13x28=17
13x=17+28
13x=45
x=4513.

Substitute x=4513 into y=5x14:
y=5(4513)14
y=2251314
y=2251318213
y=4313.

Thus, x=4513 and y=4313.


Example 96: Solve the inequality 7x+530.

Solution:
7x+530
7x305
7x25
x257
x3.57


Example 97: Solve the system of equations:
4x+3y=13
2xy=4

Solution using elimination:
Multiply the second equation by 3:
3(2xy=4)6x3y=12.

Now add both equations:
(4x+3y)+(6x3y)=13+12
10x=25
x=2510=2.5.

Substitute x=2.5 into 4x+3y=13:
4(2.5)+3y=13
10+3y=13
3y=1310
3y=3
y=1.

Thus, x=2.5 and y=1.


Example 98: Solve the inequality 5x8<17.

Solution:
5x8<17
5x<17+8
5x<25
x<255
x<5


Example 99: Solve the system of equations:
2x+5y=22
3xy=4

Solution using substitution:
From the second equation:
y=3x4.

Substitute into the first equation:
2x+5(3x4)=22
2x+15x20=22
17x20=22
17x=22+20
17x=42
x=4217.

Substitute x=4217 into y=3x4:
y=3(4217)4
y=126174
y=126176817
y=5817.

Thus, x=4217 and y=5817.


Example 100: Solve the inequality 3x+2>11.

Solution:
3x+2>11
3x>112
3x>9
x>93
x>3

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