Question

show all work!!! solve each equation 1. 8(3x - 5) - 4(x + 3) = 12 solve each equation 2. ((x - 1)/3 + x/5) = 5 solve for y 4/5+5/(y - 3)=5/(5y - 15)

Explanation:

Step1: Expand the first equation

Expand \(8(3x - 5)-4(x + 3)=12\) using the distributive property \(a(b + c)=ab+ac\). We get \(24x-40 - 4x-12 = 12\). Then combine like - terms: \((24x-4x)+(-40 - 12)=12\), which simplifies to \(20x-52 = 12\). Add 52 to both sides: \(20x=12 + 52\), so \(20x=64\). Divide both sides by 20: \(x=\frac{64}{20}=\frac{16}{5}=3.2\).

Step2: Expand the second equation

For \((\frac{x - 1}{3}+\frac{x}{5})=5\), multiply through by the least common multiple of 3 and 5, which is 15. We have \(15\times\frac{x - 1}{3}+15\times\frac{x}{5}=15\times5\). This simplifies to \(5(x - 1)+3x = 75\). Expand further: \(5x-5 + 3x=75\). Combine like - terms: \(8x-5 = 75\). Add 5 to both sides: \(8x=80\). Divide both sides by 8: \(x = 10\).

Step3: Solve the third equation for \(y\)

Given \(\frac{4}{1}+\frac{5}{y - 3}=\frac{5}{5y-15}\). First, factor the denominator on the right - hand side: \(5y-15 = 5(y - 3)\). Multiply through by \(5(y - 3)\) to clear the fractions: \(4\times5(y - 3)+5\times5 = 5\). Expand: \(20(y - 3)+25 = 5\). \(20y-60 + 25 = 5\). Combine like - terms: \(20y-35 = 5\). Add 35 to both sides: \(20y=40\). Divide both sides by 20: \(y = 2\).

Answer:

1. \(x = 3.2\)
2. \(x = 10\)
3. \(y = 2\)