Question

which equation represents the line that passes through the points (-1, -2) and (2, 10)?
a. y = 4x - 2
b. y = 3x + 1
c. y = 6x - 2
d. y = 4x - 1

Explanation:

Step1: Calculate the slope

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \). For points \((-1, -2)\) and \((3, 10)\), we have \( x_1=-1,y_1=-2,x_2 = 3,y_2=10 \). So \( m=\frac{10-(-2)}{3-(-1)}=\frac{12}{4}=3 \)? Wait, no, wait, \( 10 - (-2)=12 \), \( 3 - (-1)=4 \), \( 12/4 = 3 \)? Wait, but let's check the options. Wait, maybe I miscalculated. Wait, no, let's recalculate. Wait, the points are \((-1, -2)\) and \((3, 10)\). So \( y_2 - y_1=10 - (-2)=12 \), \( x_2 - x_1=3 - (-1)=4 \), so slope \( m = 12/4=3 \)? Wait, but the options have slopes like 4, 3, 6, 4. Wait, maybe I made a mistake. Wait, no, let's check the options again. Wait, maybe the points are \((-1, -2)\) and \((3, 10)\)? Wait, no, maybe the second point is (3,10)? Wait, let's check the slope formula again. Wait, maybe the points are \((-1, -2)\) and \((3, 10)\). Then slope \( m=(10 - (-2))/(3 - (-1))=12/4 = 3 \). But the options: first option \( y = 4x - 2 \), slope 4; second \( y=3x + 1 \), slope 3; third \( y=6x - 2 \), slope 6; fourth \( y=4x - 1 \), slope 4. Wait, maybe the points are different. Wait, maybe the points are \((-1, -2)\) and \((3, 10)\)? Wait, no, let's plug \( x=-1 \) into the options. Let's take option A: \( y = 4x - 2 \). When \( x=-1 \), \( y=4(-1)-2=-6 \), which is not -2. Option B: \( y=3x + 1 \). When \( x=-1 \), \( y=3(-1)+1=-2 \), which matches the first point. Now check the second point (3,10). Plug \( x=3 \) into B: \( y=3*3 + 1=10 \), which matches. Wait, so the slope is 3, and the equation is \( y=3x + 1 \). Wait, but earlier calculation of slope was 3, which matches option B. Wait, but let's do it properly. The slope-intercept form is \( y=mx + b \). We found \( m = 3 \). Now use point \((-1, -2)\) to find \( b \). So \( -2=3*(-1)+b \), so \( -2=-3 + b \), so \( b = 1 \). So the equation is \( y=3x + 1 \), which is option B.

Step2: Verify with the second point

For the equation \( y = 3x + 1 \), when \( x = 3 \), \( y=3*3 + 1=10 \), which matches the second point \((3, 10)\). Also, when \( x=-1 \), \( y=3*(-1)+1=-2 \), which matches the first point \((-1, -2)\). So this equation is correct.

Answer:

B. \( y = 3x + 1 \)