Solve the equation.
a. $-2(5x-1)=2x-6$
Answer (2)
To solve the equation − 2 ( 5 x − 1 ) = 2 x − 6 , distribute, combine like terms, isolate x , and solve, resulting in x = 3 2 . Verification by substituting back confirms the solution is correct. Thus, the final answer is x = 3 2 .
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Distribute -2 on the left side: − 10 x + 2 = 2 x − 6 .
Combine like terms by adding 10 x to both sides: 2 = 12 x − 6 .
Isolate the x term by adding 6 to both sides: 8 = 12 x .
Solve for x by dividing both sides by 12: x = 3 2 .
3 2
Explanation
Understanding the Problem We are given the equation − 2 ( 5 x − 1 ) = 2 x − 6 and we want to solve for x . This involves distributing, combining like terms, and isolating x .
Distributing First, distribute the − 2 on the left side of the equation: − 2 ( 5 x − 1 ) = − 10 x + 2 So the equation becomes: − 10 x + 2 = 2 x − 6
Combining Like Terms Next, we want to get all the x terms on one side of the equation. Add 10 x to both sides: − 10 x + 2 + 10 x = 2 x − 6 + 10 x 2 = 12 x − 6
Isolating x Now, add 6 to both sides to isolate the term with x : 2 + 6 = 12 x − 6 + 6 8 = 12 x
Solving for x Finally, divide both sides by 12 to solve for x : 12 8 = 12 12 x x = 12 8 Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 4: x = 12 ÷ 4 8 ÷ 4 = 3 2
Verification To verify the solution, substitute x = 3 2 back into the original equation: − 2 ( 5 ( 3 2 ) − 1 ) = 2 ( 3 2 ) − 6 − 2 ( 3 10 − 1 ) = 3 4 − 6 − 2 ( 3 10 − 3 3 ) = 3 4 − 3 18 − 2 ( 3 7 ) = − 3 14 3 − 14 = − 3 14 The solution is correct.
Examples
When solving for an unknown variable in an equation, we isolate the variable by performing inverse operations. This technique is used in various real-life scenarios, such as determining the amount of ingredients needed for a recipe when scaling it up or down, calculating the required force to move an object, or figuring out the time it takes to travel a certain distance at a given speed. Understanding how to manipulate equations is fundamental in many fields, including science, engineering, and finance.
