Question

solve each equation. use paper or the sketch tool if it helps with your thinking. \\(\frac{3y - 6}{9} = \frac{4 - 2y}{-3}\\) \\(0.3(x - 10) - 1.8 = 2.7x\\)

Explanation:

Solving \(\boldsymbol{\frac{3y - 6}{9}=\frac{4 - 2y}{-3}}\)

Step1: Cross - multiply

To eliminate the fractions, we cross - multiply. That is, if \(\frac{a}{b}=\frac{c}{d}\), then \(a\times d=b\times c\). So for \(\frac{3y - 6}{9}=\frac{4 - 2y}{-3}\), we have \((3y - 6)\times(-3)=9\times(4 - 2y)\)

Step2: Expand both sides

Expand the left - hand side: \((3y-6)\times(-3)=3y\times(-3)-6\times(-3)=-9y + 18\)
Expand the right - hand side: \(9\times(4 - 2y)=9\times4-9\times2y = 36-18y\)
So the equation becomes \(-9y + 18=36-18y\)

Step3: Add \(18y\) to both sides

Add \(18y\) to both sides of the equation: \(-9y+18y + 18=36-18y + 18y\)
Simplify: \(9y+18 = 36\)

Step4: Subtract 18 from both sides

Subtract 18 from both sides: \(9y+18 - 18=36 - 18\)
Simplify: \(9y=18\)

Step5: Divide by 9

Divide both sides by 9: \(\frac{9y}{9}=\frac{18}{9}\)
Simplify: \(y = 2\)

Solving \(\boldsymbol{0.3(x - 10)-1.8=2.7x}\)

Step1: Expand the left - hand side

Use the distributive property \(a(b - c)=ab-ac\). For \(0.3(x - 10)\), we get \(0.3x-0.3\times10=0.3x - 3\)
So the equation \(0.3(x - 10)-1.8=2.7x\) becomes \(0.3x-3-1.8=2.7x\)

Step2: Combine like terms on the left - hand side

Combine \(-3\) and \(-1.8\): \(0.3x-(3 + 1.8)=2.7x\), so \(0.3x-4.8=2.7x\)

Step3: Subtract \(0.3x\) from both sides

\(0.3x-0.3x-4.8=2.7x-0.3x\)
Simplify: \(-4.8 = 2.4x\)

Step4: Divide by 2.4

Divide both sides by 2.4: \(\frac{-4.8}{2.4}=\frac{2.4x}{2.4}\)
Simplify: \(x=-2\)

Answer:

For \(\frac{3y - 6}{9}=\frac{4 - 2y}{-3}\), \(y = 2\); for \(0.3(x - 10)-1.8=2.7x\), \(x=-2\)