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
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
For example:
is a linear equation. is another linear equation.
Steps to Solve Linear Equations:
- Simplify: Simplify both sides of the equation if necessary by combining like terms.
- Isolate the Variable: Use inverse operations (addition, subtraction, multiplication, or division) to get the variable on one side of the equation.
- Solve for the Variable: Perform the required operations to find the value of the variable.
Example: Solve
Solution:
2. What is a Linear Inequality?
A linear inequality is similar to a linear equation but involves an inequality symbol (
For example:
is a linear inequality. is another linear inequality.
Steps to Solve Linear Inequalities:
- Simplify: Simplify both sides of the inequality.
- Isolate the Variable: Use inverse operations to move the variable to one side.
- Solve for the Variable: Perform the necessary operations.
- Flip the Sign: When multiplying or dividing both sides of an inequality by a negative number, remember to flip the inequality sign.
Example: Solve
Solution:
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:
- Substitution: Solve one equation for one variable and substitute it into the other equation.
- Elimination: Add or subtract equations to eliminate one variable, then solve for the remaining variable.
- Graphing: Graph both equations and find the point where they intersect.
Example: Solve the system:
Solution using substitution:
From the first equation:
Substitute this into the second equation:
Now, substitute
Thus, the solution is
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:
- First, graph the boundary line (the equation obtained by replacing the inequality sign with an equal sign).
- Determine whether to use a solid line (for
or ) or a dashed line (for or ). - Test a point that is not on the boundary line to decide which side of the line to shade.
Example: Graph the inequality
Solution:
- Graph the boundary line
. - Since the inequality symbol is
, use a solid line. - Test the point
: , 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
Solution:
Example 2: Solve the inequality
Solution:
Example 3: Solve the system of equations:
Solution using substitution:
From the second equation:
Substitute into the first equation:
Now, substitute
Thus, the solution is
Example 4: Solve the inequality
Solution:
Example 5: Solve
Solution:
Example 6: Solve the inequality
Solution:
Example 7: Solve the system of equations:
Solution using elimination:
From the second equation, multiply by 2:
Now, add the two equations:
By adding these, we get:
Solve for
Example 8: Solve the inequality
Solution:
Example 9: Solve
Solution:
Example 10: Solve the inequality
Solution:
Examples 11 to 20
Example 11: Solve the system of equations:
Solution using substitution:
From the first equation:
Substitute into the second equation:
Now substitute
Thus,
Example 12: Solve the inequality
Solution:
Example 13: Solve
Solution:
Example 14: Solve the inequality
Solution:
Example 15: Solve the system of equations:
Solution using substitution:
From the first equation:
Substitute into the second equation:
Substitute
Thus, the solution is
Example 16: Solve the inequality
Solution:
Example 17: Solve
Solution:
Example 18: Solve the inequality
Solution:
Example 19: Solve the system of equations:
Solution using substitution:
From the second equation:
Substitute into the first equation:
Now, substitute
Thus,
Example 20: Solve the inequality
Solution:
Examples 21 to 30
Example 21: Solve the system of equations:
Solution using substitution:
From the second equation:
Substitute into the first equation:
Now substitute
Thus,
Example 22: Solve the inequality
Solution:
Example 23: Solve
Solution:
Example 24: Solve the inequality
Solution:
Example 25: Solve the system of equations:
Solution using substitution:
From the first equation:
Substitute into the second equation:
Substitute
Thus,
Example 26: Solve the inequality
Solution:
Example 27: Solve
Solution:
Example 28: Solve the inequality
Solution:
Example 29: Solve the system of equations:
Solution using elimination:
Multiply the first equation by 3 and the second equation by 2:
Subtract the second equation from the first:
Substitute
Thus,
Example 30: Solve the inequality
Solution:
Examples 31 to 40
Example 31: Solve the system of equations:
Solution using substitution:
From the first equation:
Substitute into the second equation:
Now, substitute
Thus,
Example 32: Solve the inequality
Solution:
Example 33: Solve the inequality
Solution:
Example 34: Solve the system of equations:
Solution using elimination:
Add both equations:
Substitute
Thus,
Example 35: Solve the inequality
Solution:
Example 36: Solve the system of equations:
Solution using substitution:
From the first equation:
Substitute into the second equation:
Multiply by 5 to eliminate the fraction:
Substitute
Thus,
Example 37: Solve the inequality
Solution:
Example 38: Solve the system of equations:
Solution using substitution:
From the first equation:
Substitute into the second equation:
Substitute
Thus,
Example 39: Solve the inequality
Solution:
Example 40: Solve the system of equations:
Solution using elimination:
Multiply the first equation by 3 and the second equation by 4:
Add both equations:
Now substitute
Thus,
Examples 41 to 50
Example 41: Solve the system of equations:
Solution using substitution:
From the first equation:
Substitute into the second equation:
Substitute
Thus,
Example 42: Solve the inequality
Solution:
Example 43: Solve the system of equations:
Solution using elimination:
Multiply the second equation by 3:
Now add the two equations:
Substitute
Thus,
Example 44: Solve the inequality
Solution:
Example 45: Solve the system of equations:
Solution using substitution:
From the first equation:
Substitute into the second equation:
Substitute
Thus,
Example 46: Solve the inequality
Solution:
Example 47: Solve the system of equations:
Solution using elimination:
Add both equations:
Substitute
Thus,
Example 48: Solve the inequality
Solution:
Example 49: Solve the system of equations:
Solution using substitution:
From the first equation:
Substitute into the second equation:
Substitute
Thus,
Example 50: Solve the inequality
Solution:
Examples 51 to 60
Example 51: Solve the system of equations:
Solution using substitution:
From the second equation:
Substitute into the first equation:
Substitute
Thus,
Example 52: Solve the inequality
Solution:
Example 53: Solve the system of equations:
Solution using elimination:
Add both equations:
Substitute
Thus,
Example 54: Solve the inequality
Solution:
Example 55: Solve the system of equations:
Solution using elimination:
Add both equations:
Substitute
Thus,
Example 56: Solve the inequality
Solution:
Example 57: Solve the system of equations:
Solution using substitution:
From the first equation:
Substitute into the second equation:
Multiply by 6 to eliminate the fraction:
Substitute
Thus,
Example 58: Solve the inequality
Solution:
Example 59: Solve the system of equations:
Solution using substitution:
From the first equation:
Substitute into the second equation:
Substitute
Thus,
Example 60: Solve the inequality
Solution:
Examples 61 to 70
Example 61: Solve the system of equations:
Solution using substitution:
From the first equation:
Substitute into the second equation:
Multiply by 2 to eliminate the fraction:
Substitute
Thus,
Example 62: Solve the inequality
Solution:
Example 63: Solve the system of equations:
Solution using elimination:
Add both equations:
Substitute
Thus,
Example 64: Solve the inequality
Solution:
Example 65: Solve the system of equations:
Solution using substitution:
From the first equation:
Substitute into the second equation:
Substitute
Thus,
Example 66: Solve the inequality
Solution:
Example 67: Solve the system of equations:
Solution using substitution:
From the second equation:
Substitute into the first equation:
Substitute
Thus,
Example 68: Solve the inequality
Solution:
Example 69: Solve the system of equations:
Solution using substitution:
From the second equation:
Substitute into the first equation:
Substitute
Thus,
Example 70: Solve the inequality
Solution:
Examples 71 to 80
Example 71: Solve the system of equations:
Solution using substitution:
From the second equation:
Substitute into the first equation:
Substitute
Thus,
Example 72: Solve the inequality
Solution:
Example 73: Solve the system of equations:
Solution using elimination:
Multiply the second equation by 5:
Now add both equations:
Substitute
Thus,
Example 74: Solve the inequality
Solution:
Example 75: Solve the system of equations:
Solution using substitution:
From the first equation:
Substitute into the second equation:
Multiply by 2 to eliminate the fraction:
Substitute
Thus,
Example 76: Solve the inequality
Solution:
Example 77: Solve the system of equations:
Solution using elimination:
Add both equations:
Substitute
Thus,
Example 78: Solve the inequality
Solution:
Example 79: Solve the system of equations:
Solution using substitution:
From the first equation:
Substitute into the second equation:
Substitute
Thus,
Example 80: Solve the inequality
Solution:
Examples 81 to 90
Example 81: Solve the system of equations:
Solution using substitution:
From the second equation:
Substitute into the first equation:
Substitute
Thus,
Example 82: Solve the inequality
Solution:
Example 83: Solve the system of equations:
Solution using elimination:
Multiply the second equation by 2:
Now add both equations:
Substitute
Thus,
Example 84: Solve the inequality
Solution:
Example 85: Solve the system of equations:
Solution using elimination:
Add both equations:
Substitute
Thus,
Example 86: Solve the inequality
Solution:
Example 87: Solve the system of equations:
Solution using substitution:
From the first equation:
Substitute into the second equation:
Substitute
Thus,
Example 88: Solve the inequality
Solution:
Example 89: Solve the system of equations:
Solution using substitution:
From the first equation:
Substitute into the second equation:
Substitute
Thus,
Example 90: Solve the inequality
Solution:
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
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
For example:
is a linear equation. is another linear equation.
Steps to Solve Linear Equations:
- Simplify: Simplify both sides of the equation if necessary by combining like terms.
- Isolate the Variable: Use inverse operations (addition, subtraction, multiplication, or division) to get the variable on one side of the equation.
- Solve for the Variable: Perform the required operations to find the value of the variable.
Example: Solve
Solution:
2. What is a Linear Inequality?
A linear inequality is similar to a linear equation but involves an inequality symbol (
For example:
is a linear inequality. is another linear inequality.
Steps to Solve Linear Inequalities:
- Simplify: Simplify both sides of the inequality.
- Isolate the Variable: Use inverse operations to move the variable to one side.
- Solve for the Variable: Perform the necessary operations.
- Flip the Sign: When multiplying or dividing both sides of an inequality by a negative number, remember to flip the inequality sign.
Example: Solve
Solution:
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:
- Substitution: Solve one equation for one variable and substitute it into the other equation.
- Elimination: Add or subtract equations to eliminate one variable, then solve for the remaining variable.
- Graphing: Graph both equations and find the point where they intersect.
Example: Solve the system:
Solution using substitution:
From the first equation:
Substitute this into the second equation:
Now, substitute
Thus, the solution is
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:
- First, graph the boundary line (the equation obtained by replacing the inequality sign with an equal sign).
- Determine whether to use a solid line (for
or ) or a dashed line (for or ). - Test a point that is not on the boundary line to decide which side of the line to shade.
Example: Graph the inequality
Solution:
- Graph the boundary line
. - Since the inequality symbol is
, use a solid line. - Test the point
: , 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
Solution:
Example 2: Solve the inequality
Solution:
Examples 91 to 100
Example 91: Solve the system of equations:
Solution using substitution:
From the second equation:
Substitute into the first equation:
Substitute
Thus,
Example 92: Solve the inequality
Solution:
Example 93: Solve the system of equations:
Solution using elimination:
Multiply the second equation by 3 and the first equation by 2:
Add both equations:
Substitute
Thus,
Example 94: Solve the inequality
Solution:
Example 95: Solve the system of equations:
Solution using substitution:
From the second equation:
Substitute into the first equation:
Substitute
Thus,
Example 96: Solve the inequality
Solution:
Example 97: Solve the system of equations:
Solution using elimination:
Multiply the second equation by 3:
Now add both equations:
Substitute
Thus,
Example 98: Solve the inequality
Solution:
Example 99: Solve the system of equations:
Solution using substitution:
From the second equation:
Substitute into the first equation:
Substitute
Thus,
Example 100: Solve the inequality
Solution: