Question

(a) write an equation to find x. make sure you use an \=\ sign in your answer.
equation: 15x = 10

Explanation:

Step1: Identify angle relationship

The two angles \(113^\circ\) and \((15x - 37)^\circ\) are vertical angles? No, wait, actually, when two lines intersect, adjacent angles on a straight line sum to \(180^\circ\)? Wait, no, looking at the diagram, the angle \(113^\circ\) and \((15x - 37)^\circ\) should be equal? Wait, no, maybe they are supplementary? Wait, no, let's re - examine. Wait, the angle \(113^\circ\) and \((15x - 37)^\circ\) and the other angle. Wait, actually, if we consider that the sum of angles around a point or on a straight line. Wait, maybe the angle \(113^\circ\) and \((15x - 37)^\circ\) are equal? No, wait, maybe the angle \(113^\circ\) and \((15x - 37)^\circ\) are supplementary? Wait, no, let's think again. Wait, the correct relationship: when two lines intersect, vertical angles are equal, and adjacent angles are supplementary. Wait, in the diagram, the angle \(113^\circ\) and \((15x - 37)^\circ\) are vertical angles? No, maybe the angle \(113^\circ\) and \((15x - 37)^\circ\) are equal? Wait, no, let's calculate. Wait, if we assume that \(113=(15x - 37)+ \text{something}\)? No, wait, the problem is to write an equation to find \(x\). Wait, maybe the angle \(113^\circ\) and \((15x - 37)^\circ\) are supplementary? Wait, no, \(113+(15x - 37)=180\)? Wait, \(113 - 37+15x=180\), \(76 + 15x=180\), \(15x=180 - 76=104\)? No, that's not matching the given equation. Wait, the given equation in the box is \(15x = 10\), but that seems wrong. Wait, maybe I misread. Wait, the angle \(113^\circ\) and \((15x - 37)^\circ\) are equal? So \(113=15x - 37\). Then, adding 37 to both sides: \(113 + 37=15x\), \(150 = 15x\), \(x = 10\). Oh! So the correct equation is \(15x-37 = 113\), then \(15x=113 + 37=150\), but the given equation in the box is \(15x = 10\), which is wrong. Wait, no, maybe the diagram is different. Wait, maybe the angle \(113^\circ\) and \((15x - 37)^\circ\) have a relationship where \(113-(15x - 37)=0\)? No, let's do it properly.

If the angle \(113^\circ\) and \((15x - 37)^\circ\) are equal (vertical angles), then:

Step1: Set up the equation

Since vertical angles are equal, we have \(15x-37 = 113\)

Step2: Solve for \(15x\)

Add 37 to both sides of the equation \(15x-37=113\)
\(15x-37 + 37=113 + 37\)
\(15x=150\)

But the given equation in the box is \(15x = 10\), which is incorrect. Wait, maybe the diagram is such that the angle \(113^\circ\) and \((15x - 37)^\circ\) are supplementary? No, \(113+(15x - 37)=180\), \(15x+76 = 180\), \(15x=104\), which is not. Wait, maybe the angle \(113^\circ\) is equal to \((15x - 37)^\circ\) plus some angle, but the user's diagram shows that the equation is \(15x = 10\), which is wrong. Wait, no, maybe I made a mistake. Wait, the correct equation should be derived from the angle relationship. If we assume that the angle \((15x - 37)^\circ\) and \(113^\circ\) are related such that \(15x-37=113\), then \(15x=113 + 37=150\), \(x = 10\). So the equation to find \(x\) is \(15x-37=113\), and when we solve it, we get \(15x=150\), but the given box has \(15x = 10\), which is incorrect. But according to the problem, we need to write the equation. Wait, maybe the angle \(113^\circ\) and \((15x - 37)^\circ\) are equal, so the equation is \(15x-37 = 113\), and then \(15x=113 + 37=150\), but the user's box has \(15x = 10\), which is wrong. But maybe the problem has a typo, but according to the correct angle relationship (vertical angles are equal), the equation is \(15x-37=113\), and solving it gives \(15x = 150\), \(x = 10\).

Answer:

The correct equation to find \(x\) is \(15x - 37=113\) (and the solution for \(x\) is \(x = 10\) since \(15x=150\) when we add 37 to both sides of the equation \(15x - 37=113\)). But if we follow the given incorrect equation \(15x = 10\), \(x=\frac{10}{15}=\frac{2}{3}\), but that is wrong. The correct equation is \(15x-37 = 113\) leading to \(x = 10\).