1. ) [tex]$6 x+5=18-7 x$[/tex]
2. ) [tex]$6+8 x=3 x+6 x$[/tex]
3. ) [tex]$7 x+2=4 x-7$[/tex]
Answer (1)
Solve the first equation 6 x + 5 = 18 − 7 x and find x = 1 .
Solve the second equation 6 + 8 x = 3 x + 6 x and find x = 6 .
Solve the third equation 7 x + 2 = 4 x − 7 and find x = − 3 .
The solutions are x = 1 , 6 , − 3 .
Explanation
Problem Analysis We are given three linear equations and our goal is to solve each one for the variable x . We will use algebraic manipulation to isolate x on one side of each equation.
Solving Equation 1 Let's solve the first equation: 6 x + 5 = 18 − 7 x .
First, add 7 x to both sides of the equation: 6 x + 7 x + 5 = 18 − 7 x + 7 x
13 x + 5 = 18
Next, subtract 5 from both sides: 13 x + 5 − 5 = 18 − 5 13 x = 13
Finally, divide both sides by 13: 13 13 x = 13 13 x = 1
Solving Equation 2 Now, let's solve the second equation: 6 + 8 x = 3 x + 6 x .
Combine like terms on the right side: 6 + 8 x = 9 x
Subtract 8 x from both sides: 6 + 8 x − 8 x = 9 x − 8 x 6 = x So, x = 6 .
Solving Equation 3 Let's solve the third equation: 7 x + 2 = 4 x − 7 .
Subtract 4 x from both sides: 7 x − 4 x + 2 = 4 x − 4 x − 7 3 x + 2 = − 7
Subtract 2 from both sides: 3 x + 2 − 2 = − 7 − 2 3 x = − 9
Divide both sides by 3: 3 3 x = 3 − 9 x = − 3
Final Answer Therefore, the solutions to the three equations are:
x = 1
x = 6
x = − 3
Examples
Linear equations are used in various real-life scenarios, such as calculating the cost of items, determining the speed of a vehicle, or converting between different units of measurement. For example, if you know the hourly rate of a job and the total amount earned, you can use a linear equation to find the number of hours worked. Understanding how to solve linear equations is a fundamental skill that can be applied in many practical situations.
