Question

1. find the values of x and y. (18y + 5)° (10x - 61)° (x + 10)°

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. So, \(10x−61=x + 10\).

Step2: Solve for \(x\)

Subtract \(x\) from both sides: \(10x−x−61=x−x + 10\), which simplifies to \(9x−61 = 10\). Then add 61 to both sides: \(9x−61 + 61=10 + 61\), getting \(9x=71\), and \(x=\frac{71}{9}\approx7.89\).

Step3: Use linear - pair property

The angles \((18y + 5)\) and \((10x−61)\) form a linear - pair, so \((18y + 5)+(10x−61)=180\). Substitute \(x = \frac{71}{9}\) into the equation: \(18y+5+(10\times\frac{71}{9}-61)=180\). First, calculate \(10\times\frac{71}{9}-61=\frac{710}{9}-\frac{549}{9}=\frac{710 - 549}{9}=\frac{161}{9}\). Then the equation becomes \(18y+5+\frac{161}{9}=180\). Multiply through by 9 to clear the fraction: \(162y+45 + 161=1620\). Combine like - terms: \(162y+206 = 1620\). Subtract 206 from both sides: \(162y=1620 - 206=1414\). So, \(y=\frac{1414}{162}=\frac{707}{81}\approx8.73\).

Answer:

\(x=\frac{71}{9}\), \(y = \frac{707}{81}\)