Question

monicas teacher looked at her work for creating an equation of a line in slope - intercept form given 2 coordinates, and she informed her that there was an error in her calculations. identify which step the error first occurred. (18, 18) and (14, 14) step 1: $m = \frac{14 - 18}{14 - 18} = \frac{-4}{-4} = 1$ step 2: $y - 14 = 1(x - 18)$ step 3: $y - 14 = x - 18$ step 4: $y = x - 4$ options: step 2, step 1, step 3, step 4

Explanation:

Step1: Analyze Step 1 (Slope Calculation)

The slope formula is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). For points \((18, 18)\) (let \( (x_1,y_1)=(18,18) \)) and \((14, 14)\) (let \( (x_2,y_2)=(14,14) \)), the slope should be \( m=\frac{14 - 18}{14 - 18}=\frac{-4}{-4} = 1 \). Wait, but actually, when using the two points, if we take \((x_1,y_1)=(18,18)\) and \((x_2,y_2)=(14,14)\), the slope is correct here? Wait no, wait Step 2: The point - slope form is \( y - y_1=m(x - x_1) \). In Step 2, she used \( y - 14 = 1(x - 18) \), but if \( (x_1,y_1) \) is \((18,18)\), then it should be \( y - 18=1(x - 18) \), or if \( (x_2,y_2)=(14,14) \), it should be \( y - 14=1(x - 14) \). Wait, let's re - check Step 1 first. Wait the slope calculation: \( m=\frac{14 - 18}{14 - 18} \), but the denominator should be \( x_2 - x_1=14 - 18 \) and numerator \( y_2 - y_1 = 14 - 18 \), so that's correct, \( \frac{-4}{-4}=1 \). Now Step 2: The point - slope formula is \( y - y_1=m(x - x_1) \). She used the point \((18,18)\) as \( x_1,y_1 \)? No, she wrote \( y - 14=1(x - 18) \). So if the point is \((14,14)\), then \( x_1 = 14,y_1 = 14 \), so the equation should be \( y - 14=1(x - 14) \), not \( y - 14=1(x - 18) \). Wait, but let's check the steps in order. Wait Step 1: slope is calculated as \( \frac{14 - 18}{14 - 18} \), but actually, the slope formula is \( \frac{y_2 - y_1}{x_2 - x_1} \). Let's take the two points \((18,18)\) and \((14,14)\). Let \( (x_1,y_1)=(18,18) \) and \( (x_2,y_2)=(14,14) \). Then \( m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{14 - 18}{14 - 18}=\frac{-4}{-4}=1 \), that's correct. Now Step 2: The point - slope form. If we use the point \((18,18)\), the equation is \( y - 18=m(x - 18) \). If we use the point \((14,14)\), the equation is \( y - 14=m(x - 14) \). But in Step 2, she used \( y - 14 = 1(x - 18) \), which is mixing the \( x \) from \((18,18)\) and \( y \) from \((14,14)\). So Step 2 has an error. Wait, but let's check again. Wait the problem is to find where the error first occurred. Let's go step by step.

Step 1: Slope calculation. \( m=\frac{14 - 18}{14 - 18}=\frac{-4}{-4}=1 \). That's correct because \( \frac{-4}{-4}=1 \).

Step 2: Point - slope form. The point - slope formula is \( y - y_1=m(x - x_1) \), where \((x_1,y_1)\) is a point on the line. She has two points: \((18,18)\) and \((14,14)\). If she uses the point \((14,14)\) (so \( x_1 = 14,y_1 = 14 \)), then the equation should be \( y - 14=1(x - 14) \), but she wrote \( y - 14=1(x - 18) \). If she uses the point \((18,18)\) (so \( x_1 = 18,y_1 = 18 \)), the equation should be \( y - 18=1(x - 18) \). So in Step 2, she made a mistake in the point - slope form by using \( x = 18 \) with \( y = 14 \).

Wait, but let's check the options. The options are Step 2, Step 1, Step 3, Step 4. Wait, maybe I made a mistake in Step 1. Wait the slope calculation: \( \frac{14 - 18}{14 - 18} \), but the denominator is \( x_2 - x_1=14 - 18 \) and numerator \( y_2 - y_1 = 14 - 18 \), so that's correct. Then Step 2: The point - slope form. She used \( y - 14=1(x - 18) \), but the correct point - slope form with \( m = 1 \) and point \((14,14)\) is \( y - 14=1(x - 14) \), or with point \((18,18)\) is \( y - 18=1(x - 18) \). So Step 2 has an error. But wait, let's check the original problem again. Wait the two points are \((18,18)\) and \((14,14)\). Let's recalculate the slope: \( m=\frac{14 - 18}{14 - 18}=\frac{-4}{-4}=1 \), that's correct. Then Step 2: The point - slope equation. If we take the point \((18,18)\), then \( y - 18=1(x - 18) \), which simplifies to \( y=x \). If we take the point \((14,14)\), \( y - 14=1(x - 14) \), which also simplifies to \( y=x \). But in Step 2, she wrote \( y - 14=1(x - 18) \), which is wrong. So the error first occurs in Step 2? Wait no, wait maybe I messed up. Wait let's check the steps again:

Step 1: Slope \( m=\frac{14 - 18}{14 - 18}=\frac{-4}{-4}=1 \). Correct.

Step 2: Point - slope form. The formula is \( y - y_1=m(x - x_1) \). She used \( y - 14 \) (so \( y_1 = 14 \)) and \( x - 18 \) (so \( x_1 = 18 \)). But \( x_1 \) and \( y_1 \) should be from the same point. So either \( (x_1,y_1)=(18,18) \) (so \( y - 18=1(x - 18) \)) or \( (x_1,y_1)=(14,14) \) (so \( y - 14=1(x - 14) \)). So Step 2 is incorrect because she mixed the \( x \) and \( y \) from different points.

But wait, the options include Step 2. Wait but let's check the other steps. Step 3: \( y - 14=x - 18 \), which is expanding Step 2. Step 4: \( y=x - 4 \), which is solving Step 3 for \( y \). But the error starts in Step 2. Wait, but maybe I made a mistake in Step 1. Wait the slope formula: \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Let's take \( (x_1,y_1)=(18,18) \) and \( (x_2,y_2)=(14,14) \). Then \( m=\frac{14 - 18}{14 - 18}=\frac{-4}{-4}=1 \). Correct. So Step 1 is correct. Step 2: Incorrect, because of the point - slope form. So the error first occurs in Step 2? Wait no, wait the problem says "identify which step the error first occurred". Let's re - examine:

Wait, in Step 2, the point - slope equation is written as \( y - 14 = 1(x - 18) \). The correct point - slope equation using the slope \( m = 1 \) and one of the points:

- Using \((18,18)\): \( y - 18=1(x - 18) \)
- Using \((14,14)\): \( y - 14=1(x - 14) \)

So Step 2 is incorrect. So the answer should be Step 2? Wait but let's check the options. The options are Step 2, Step 1, Step 3, Step 4. Wait, maybe I made a mistake in Step 1. Wait the slope calculation: \( \frac{14 - 18}{14 - 18} \), but the denominator is \( x_2 - x_1=14 - 18 \) and numerator \( y_2 - y_1 = 14 - 18 \), so that's correct. So Step 1 is correct. Step 2 is incorrect. So the error first occurs in Step 2.

Wait, but let's check the original problem again. The two points are \((18,18)\) and \((14,14)\). Let's calculate the slope again: \( m=\frac{14 - 18}{14 - 18}=\frac{-4}{-4}=1 \), correct. Then Step 2: The point - slope form. She used \( y - 14=1(x - 18) \), which is wrong. So the error is in Step 2.

Step 2: Analyze Step 2 (Point - Slope Form)

The point - slope formula is \( y - y_1=m(x - x_1) \), where \((x_1,y_1)\) is a point on the line. For the points \((18,18)\) and \((14,14)\) with \( m = 1 \):
- If we use \((x_1,y_1)=(18,18)\), the equation is \( y - 18=1(x - 18) \).
- If we use \((x_1,y_1)=(14,14)\), the equation is \( y - 14=1(x - 14) \).

In Step 2, Monica used \( y - 14=1(x - 18) \), which mixes the \( x \) - coordinate from \((18,18)\) and the \( y \) - coordinate from \((14,14)\). This is an error. Since Step 1 was correct (slope calculation), the first error occurs in Step 2.

Answer:

Step 2