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Chapter 4: Problem 59

Solve each equation. Check the solution. $$ 4 x-2(1-3 x)=6 $$

Short Answer

Expert verified
x = 0.8

Step by step solution

01

Distribute

Distribute the -2 across the terms inside the parentheses.Equation: 4x - 2(1 - 3x) = 6Distributed: 4x - 2 + 6x = 6
02

Combine Like Terms

Combine the terms with x to simplify the equation. 4x + 6x - 2 = 6 Simplified equation: 10x - 2 = 6
03

Isolate the Variable Term

Add 2 to both sides of the equation to get the variable term by itself.10x - 2 + 2 = 6 + 2 Simplified equation: 10x = 8
04

Solve for x

Divide both sides by 10 to solve for x. 10x / 10 = 8 / 10 Simplified equation: x = 0.8
05

Check the solution

Substitute 0.8 into the original equation to verify the solution.4(0.8) - 2(1 - 3(0.8)) = 6 Simplify inside the parentheses first:4(0.8) - 2(1 - 2.4) Then simplify further: 4(0.8) - 2(-1.4) Evaluate the multiplication operations: 3.2 + 2.8 = 6 Confirming that 6 = 6, which holds true. Therefore, x = 0.8 is the correct solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
When solving linear equations, the distributive property helps to eliminate parentheses. This property states that for any numbers a, b, and c: a(b + c) = ab + acIn our exercise, we start with the equation: 4x - 2(1 - 3x) = 6We distribute the -2 to both 1 and -3x: 4x - 2 * 1 + 2 * 3x = 6This simplifies to: 4x - 2 + 6x = 6
Combining Like Terms
After applying the distributive property, we move to combine like terms. Like terms are terms that have the same variable raised to the same power.In the equation: 4x - 2 + 6x = 64x and 6x are like terms. We combine them: 4x + 6x = 10xSo, the equation simplifies to: 10x - 2 = 6
Isolating the Variable
Next, we isolate the variable to solve for x. This means getting x alone on one side of the equation. Starting with: 10x - 2 = 6We add 2 to both sides to remove the -2: 10x - 2 + 2 = 6 + 2 This simplifies to: 10x = 8To solve for x, we divide both sides by 10: 10x/10 = 8/10This gives us: x = 0.8
Checking the Solution
Finally, we check our solution by substituting x = 0.8 back into the original equation. This verifies our answer. Original equation: 4x - 2(1 - 3x) = 6Substituting x = 0.8: 4(0.8) - 2(1 - 3(0.8))First, calculate inside the parentheses: 4(0.8) - 2(1 - 2.4)Simplifies to: 4(0.8) - 2(-1.4)Then evaluate the multiplications: 3.2 + 2.8This equals 6, confirming our solution is correct: 6 = 6Thus, x = 0.8 is the correct solution.

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