Question

given: $14 - 2(x + 8) = 5x - (3x - 34)$; prove: $x = -9$

Explanation:

Step1: Expand both sides

Expand the left - hand side: \(14-2(x + 8)=14-2x-16\) (using the distributive property \(a(b + c)=ab+ac\), here \(a=-2\), \(b = x\), \(c = 8\)). Expand the right - hand side: \(5x-(3x - 34)=5x-3x + 34\) (using the distributive property \(a-(b - c)=a - b + c\), here \(a = 5x\), \(b=3x\), \(c = 34\)).

Step2: Simplify both sides

Simplify the left - hand side: \(14-2x-16=-2x - 2\). Simplify the right - hand side: \(5x-3x + 34 = 2x+34\). So the equation becomes \(-2x-2=2x + 34\).

Step3: Add \(2x\) to both sides

Add \(2x\) to both sides of the equation \(-2x-2=2x + 34\): \(-2x-2+2x=2x + 34+2x\) (addition property of equality: if \(a=b\), then \(a + c=b + c\)). This simplifies to \(-2=4x + 34\).

Step4: Subtract 34 from both sides

Subtract 34 from both sides: \(-2-34=4x+34 - 34\) (subtraction property of equality: if \(a=b\), then \(a - c=b - c\)). This gives \(-36 = 4x\).

Step5: Divide both sides by 4

Divide both sides by 4: \(\frac{-36}{4}=\frac{4x}{4}\) (division property of equality: if \(a=b\) and \(c
eq0\), then \(\frac{a}{c}=\frac{b}{c}\)). This simplifies to \(x=-9\).

Answer:

By following the steps of expanding, simplifying, and using the properties of equality (addition, subtraction, division), we have shown that for the equation \(14 - 2(x + 8)=5x-(3x - 34)\), \(x=-9\).