Question

6 find the value of y. 9x + 42 = 15x -9x -9x 42 = 6x 6 6 x = 7 (15x)° (9x + 42)° (4y - 13)°

Explanation:

Step1: Find the measure of the vertical angle

First, we know that \(9x + 42\) and \(15x\) are vertical angles? Wait, no, actually, from the diagram, \(9x + 42\) and \(15x\) are adjacent angles forming a linear pair? Wait, no, the work shown already solved for \(x\) as \(x = 7\). Let's check: if \(x = 7\), then \(15x=15\times7 = 105^{\circ}\), and \(9x + 42=9\times7+42 = 63 + 42=105^{\circ}\), so they are equal, meaning they are vertical angles (since vertical angles are equal). Now, the angle \(15x\) and the angle \(4y - 13\) are related? Wait, looking at the diagram, the two lines (the one with \(15x\) and the other with \(4y - 13\)) seem to be parallel? Wait, no, maybe the angle \(15x\) and \(4y - 13\) are equal? Wait, no, let's see: if the two lines are parallel, then corresponding angles are equal. Wait, the angle \(15x\) (which is \(105^{\circ}\)) and the angle \(4y - 13\) – wait, maybe the angle \(15x\) and \(4y - 13\) are equal? Wait, no, let's check the diagram again. Wait, the angle \(9x + 42\) and \(15x\) are vertical angles (since they are opposite each other when two lines intersect), so they are equal, which we verified. Now, the other angle, \(4y - 13\), is equal to \(9x + 42\) or \(15x\) because of parallel lines? Wait, the two lines (the one with the arrow and the one with \(4y - 13\)) – maybe they are parallel, so the corresponding angles are equal. So \(15x=4y - 13\)? Wait, no, wait, \(15x\) is \(105^{\circ}\), so let's substitute \(x = 7\) into \(15x\), we get \(15\times7 = 105\). Then, since the angle \(15x\) and \(4y - 13\) are equal (because of parallel lines, corresponding angles), so:

Step2: Solve for \(y\)

Set \(15x=4y - 13\). We know \(x = 7\), so \(15\times7=4y - 13\)

\(105 = 4y - 13\)

Add 13 to both sides: \(105 + 13=4y\)

\(118 = 4y\)? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, \(9x + 42 = 15x\) gives \(x = 7\), so \(15x = 105\). Now, looking at the diagram, the angle \(4y - 13\) and the angle \(9x + 42\) – wait, maybe they are alternate interior angles? Wait, no, maybe the angle \(4y - 13\) is equal to \(180 - 15x\)? Wait, no, let's check the diagram again. Wait, the two lines (the transversal and the two parallel lines) – maybe the angle \(4y - 13\) is equal to \(9x + 42\) because they are corresponding angles. Wait, \(9x + 42 = 105\), so \(4y - 13 = 105\)? Wait, no, \(4y - 13 = 105\) would give \(4y = 118\), \(y = 29.5\), which is not an integer. Wait, maybe I messed up the angle relationship. Wait, the angle \(15x\) and \(4y - 13\) – maybe they are complementary? No, that doesn't make sense. Wait, wait, the work shown already found \(x = 7\), so \(15x = 105\). Now, the angle \(4y - 13\) and the angle \(15x\) – maybe they are equal because of vertical angles or parallel lines. Wait, let's re-express:

Wait, the two angles \(9x + 42\) and \(15x\) are vertical angles, so they are equal, so \(x = 7\) is correct. Now, the angle \(4y - 13\) and the angle \(9x + 42\) – wait, maybe they are equal because the lines are parallel, so corresponding angles. So \(9x + 42 = 4y - 13\)? Wait, \(9x + 42 = 105\), so \(105 = 4y - 13\), then \(4y = 118\), \(y = 29.5\). But that seems odd. Wait, maybe the angle \(4y - 13\) is equal to \(180 - 15x\)? Let's check: \(180 - 105 = 75\), so \(4y - 13 = 75\), then \(4y = 88\), \(y = 22\). Ah, that makes sense. So maybe the angle \(15x\) and \(4y - 13\) are supplementary? Wait, no, if the two lines are parallel, then consecutive interior angles are supplementary. Wait, let's look at the diagram again. The transversal crosses two parallel lines. The angle \(15x\) is on one line, and \(4y - 13\) is on the other line, same side of the transversal, so they are consecutive interior angles, hence supplementary. So \(15x + (4y - 13) = 180\). Let's verify:

\(15x = 105\), so \(105 + 4y - 13 = 180\)

\(92 + 4y = 180\)

\(4y = 88\)

\(y = 22\)

Yes, that works. So the correct relationship is that \(15x\) and \(4y - 13\) are supplementary (consecutive interior angles) because the lines are parallel. So:

Step1: Calculate \(15x\)

Given \(x = 7\), so \(15x = 15\times7 = 105\)

Step2: Set up the supplementary angle equation

Since \(15x\) and \(4y - 13\) are supplementary, their sum is \(180^{\circ}\):

\(15x + (4y - 13) = 180\)

Substitute \(15x = 105\):

\(105 + 4y - 13 = 180\)

Step3: Simplify and solve for \(y\)

Simplify the left side: \(92 + 4y = 180\)

Subtract 92 from both sides: \(4y = 180 - 92 = 88\)

Divide both sides by 4: \(y = \frac{88}{4} = 22\)

Answer:

\(y = 22\)