Question

lets focus on x first. visually, the x parts of the diagram shows consecutive angles. this means that the angles are supplementary, which means that the two parts add up to 180 (part + part = 180). to solve for x, a good equation to set up is therefore, and therefore x = 21. now lets do the y part. suppose i choose to work with the 6x + 7 expression. (its a little bit easier that way.) using the value of x you calculated, what is the value of 6x + 7? visually, the relationship between the 6x + 7 and the 11y - 32 in the diagram is vertical angles. this means that the angles are congruent, which means that the two parts are equal to each

Explanation:

Step1: Solve for \(x\)

Since the angles are supplementary (add up to \(180^{\circ}\)), the equation is \((6x + 7)+(11y-32)=180\). But first, if we assume the other angle (not shown in the \(y\) part initially) is \(180-(6x + 7)\). But from the given \(x\) - related part (assuming the two angles involving \(x\) - related setup, if we consider the two adjacent angles (one of them is \(6x+7\) and the other is \(180-(6x + 7)\) which is not shown fully. But if we assume the equation for \(x\) (maybe a linear pair): \(6x+7+(180-(6x + 7)) = 180\) is wrong. Wait, no, if we consider that the two angles (one is \(6x + 7\) and the other adjacent angle (not involving \(y\)) sum to \(180\). But if we use \(x = 21\), then \(6x+7=6\times21 + 7=126+7 = 133\)

Step2: Solve for \(y\)

Since \(6x + 7\) and \(11y-32\) are vertical angles (congruent), we set \(6x+7=11y - 32\). Substitute \(x = 21\) into the equation: \(133=11y-32\). Add \(32\) to both sides: \(133 + 32=11y\), so \(165=11y\). Divide both sides by \(11\): \(y=\frac{165}{11}=15\)

Answer:

The value of \(6x + 7\) when \(x = 21\) is \(133\) and \(y = 15\)