Solve each equation.
$-12(-9 x-4)-5=0$
Answer: $x=$ $\square$
$12(x-6)-8(x-3)=-7$
Answer: $x=$ $\square$
Answer (1)
Distribute and simplify the first equation: − 12 ( − 9 x − 4 ) − 5 = 0 becomes 108 x − 53 = 0 .
Solve for x in the first equation: x = 108 53 .
Distribute and simplify the second equation: 12 ( x − 6 ) − 8 ( x − 3 ) = − 7 becomes 4 x − 48 = − 7 .
Solve for x in the second equation: x = 4 41 .
Explanation
Introduction We are given two equations to solve for x . Let's solve them one by one.
Solving the First Equation The first equation is − 12 ( − 9 x − 4 ) − 5 = 0 . To solve for x , we first distribute the − 12 into the parentheses: − 12 ( − 9 x ) − 12 ( 4 ) − 5 = 0
108 x − 48 − 5 = 0
108 x − 53 = 0
Now, we add 53 to both sides of the equation: 108 x = 53
Finally, we divide both sides by 108 to isolate x : x = 108 53 .
Solving the Second Equation The second equation is 12 ( x − 6 ) − 8 ( x − 3 ) = − 7 . To solve for x , we first distribute the 12 and -8 into the parentheses: 12 x − 12 ( 6 ) − 8 x + 8 ( 3 ) = − 7
12 x − 72 − 8 x + 24 = − 7
Now, we combine like terms: 4 x − 48 = − 7
Next, we add 48 to both sides of the equation: 4 x = 41
Finally, we divide both sides by 4 to isolate x : x = 4 41 .
Final Answer Therefore, the solutions for the two equations are x = 108 53 and x = 4 41 .
Examples
Solving linear equations is a fundamental skill in algebra and is used in various real-life situations. For example, if you are trying to determine how many hours you need to work to earn a certain amount of money, you can set up a linear equation to represent the situation and solve for the number of hours. Similarly, if you are calculating the cost of a project with fixed and variable expenses, you can use linear equations to model the costs and determine the total expense.
