Question

-4x + 51 = 2x + 7(x + 1)
gallons of water and is
gallons per minute. a
102.2 gallons of water and
3.5 gallons per minute.
when the pools contain
water?

1. the square and

have equal perimete
then, calculate the

Explanation:

Step1: Expand the right - hand side

First, we need to expand the expression \(7(x + 1)\) on the right - hand side of the equation \(-4x+51 = 2x+7(x + 1)\). According to the distributive property \(a(b + c)=ab+ac\), where \(a = 7\), \(b=x\) and \(c = 1\), we have \(7(x + 1)=7x+7\). So the equation becomes \(-4x + 51=2x+7x + 7\).

Step2: Combine like terms on the right - hand side

Combine the \(x\) terms on the right - hand side. \(2x+7x=9x\), so the equation is now \(-4x + 51=9x+7\).

Step3: Move the \(x\) terms to one side

Add \(4x\) to both sides of the equation to get all the \(x\) terms on the right - hand side. \(-4x+4x + 51=9x+4x+7\), which simplifies to \(51 = 13x+7\).

Step4: Isolate the constant term on the left - hand side

Subtract 7 from both sides of the equation. \(51-7=13x+7 - 7\), so \(44 = 13x\). Wait, no, wait, let's correct that. Wait, \(51-7 = 44\), so \(44=13x\)? Wait, no, let's do it again. Wait, from \(51=13x + 7\), subtract 7 from both sides: \(51-7=13x\), \(44 = 13x\)? Wait, that can't be right. Wait, no, I made a mistake in step 3. Let's go back.

Wait, the original equation is \(-4x + 51=2x+7(x + 1)\). Expand the right - hand side: \(2x+7x + 7=9x + 7\). Then, we can add \(4x\) to both sides: \(-4x+4x+51=9x + 4x+7\), so \(51=13x + 7\). Then subtract 7 from both sides: \(51 - 7=13x\), \(44 = 13x\)? Wait, that gives \(x=\frac{44}{13}\approx3.38\), but that seems odd. Wait, maybe I made a mistake in the sign. Let's try moving the \(9x\) to the left - hand side instead. Subtract \(9x\) from both sides: \(-4x-9x + 51=9x-9x+7\), so \(-13x + 51=7\). Then subtract 51 from both sides: \(-13x=7 - 51=-44\). Then divide both sides by \(-13\): \(x=\frac{-44}{-13}=\frac{44}{13}\approx3.38\). Wait, maybe the original equation was written incorrectly? Wait, let's check the original problem again. The equation is \(-4x + 51=2x+7(x + 1)\). Let's verify the expansion again. \(7(x + 1)=7x + 7\), so right - hand side is \(2x+7x + 7 = 9x+7\). Left - hand side is \(-4x + 51\). So, \(-4x+51=9x + 7\). Then, add \(4x\) to both sides: \(51=13x + 7\). Subtract 7: \(44 = 13x\), so \(x=\frac{44}{13}\).

Wait, maybe I misread the equation. Let's check the original image. The equation is \(-4x + 51=2x+7(x + 1)\)? Wait, maybe it's \(-4x + 51=2x+7(x - 1)\)? If that's the case, let's re - solve.

If the equation is \(-4x + 51=2x+7(x - 1)\), expand the right - hand side: \(2x+7x-7=9x - 7\). Then the equation is \(-4x + 51=9x-7\). Add \(4x\) to both sides: \(51=13x-7\). Add 7 to both sides: \(58 = 13x\), \(x=\frac{58}{13}\approx4.46\). But since the user provided the equation as \(-4x + 51=2x+7(x + 1)\), we will go with that.

Wait, let's do the calculation again.

Starting with \(-4x + 51=2x+7(x + 1)\)

Step1: Expand \(7(x + 1)\)

Using the distributive property \(a(b + c)=ab+ac\), \(7(x + 1)=7x+7\). So the equation is \(-4x + 51=2x+7x + 7\)

Step2: Combine like terms on the right

\(2x+7x = 9x\), so \(-4x + 51=9x+7\)

Step3: Move \(x\) terms to one side

Add \(4x\) to both sides: \(51=13x + 7\)

Step4: Solve for \(x\)

Subtract 7 from both sides: \(51 - 7=13x\), \(44 = 13x\)

Then \(x=\frac{44}{13}\approx3.38\)

Answer:

\(x = \frac{44}{13}\) (or approximately \(x\approx3.38\))