Question

write the equation of this line in slope-intercept form.
write your answer using integers, proper fractions, and improper fractions in simplest form.

Explanation:

Step1: Identify slope-intercept form

Slope - intercept form is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y - intercept.

Step2: Find the y - intercept (\( b \))

The line crosses the y - axis at \( (0,2) \), so \( b = 2 \).

Step3: Calculate the slope (\( m \))

We can use two points on the line, e.g., \( (0,2) \) and \( (- 3,0) \). The slope formula is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Let \( (x_1,y_1)=(0,2) \) and \( (x_2,y_2)=(-3,0) \). Then \( m=\frac{0 - 2}{-3 - 0}=\frac{-2}{-3}=\frac{2}{3} \). Wait, no, let's take another pair. Let's take \( (0,2) \) and \( (3,4) \). Then \( m=\frac{4 - 2}{3 - 0}=\frac{2}{3} \)? Wait, no, looking at the graph, when \( x = 3 \), \( y=4 \)? Wait, no, the y - intercept is 2, and when \( x = 3 \), let's check the grid. Wait, the line passes through \( (0,2) \) and \( (3,4) \)? Wait, no, maybe I made a mistake. Wait, the line passes through \( (0,2) \) and \( (-3,0) \). Wait, \( \frac{2 - 0}{0-(-3)}=\frac{2}{3} \)? Wait, no, slope is rise over run. From \( (-3,0) \) to \( (0,2) \), the rise is \( 2-0 = 2 \), the run is \( 0-(-3)=3 \), so slope \( m=\frac{2}{3} \)? Wait, no, wait the line: when \( x = 0 \), \( y = 2 \); when \( x = 3 \), \( y = 4 \)? Wait, no, the grid lines: each square is 1 unit. Let's take two clear points. The line passes through \( (0,2) \) and \( (3,4) \)? Wait, no, maybe \( (0,2) \) and \( ( - 3,0) \). Wait, the slope formula: \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Let's take \( (x_1,y_1)=(0,2) \) and \( (x_2,y_2)=(3,4) \). Then \( m=\frac{4 - 2}{3 - 0}=\frac{2}{3} \)? Wait, no, that can't be. Wait, maybe I misread the graph. Wait, the line: when \( x = 0 \), \( y = 2 \); when \( x = 3 \), \( y = 4 \)? Wait, no, let's count the rise and run. From \( (0,2) \) to \( (3,4) \), rise is 2, run is 3, so slope is \( \frac{2}{3} \)? Wait, but let's check another point. When \( x=-3 \), \( y = 0 \). So from \( (-3,0) \) to \( (0,2) \), rise is \( 2-0 = 2 \), run is \( 0 - (-3)=3 \), so slope is \( \frac{2}{3} \). Then the equation is \( y=\frac{2}{3}x + 2 \)? Wait, no, wait, maybe I made a mistake. Wait, let's take \( (0,2) \) and \( (3,4) \): \( y - 2=\frac{4 - 2}{3 - 0}(x - 0)\), so \( y-2=\frac{2}{3}x \), so \( y=\frac{2}{3}x+2 \). Wait, but let's check \( x = - 3 \): \( y=\frac{2}{3}(-3)+2=-2 + 2=0 \), which matches the point \( (-3,0) \). So that's correct.

Wait, no, wait the original graph: the line goes through \( (0,2) \) and when \( x = 3 \), \( y = 4 \), so slope is \( \frac{2}{3} \), y - intercept is 2. So the equation is \( y=\frac{2}{3}x + 2 \)? Wait, no, wait maybe I messed up the points. Wait, let's look again. The line passes through \( (0,2) \) and \( (3,4) \), so slope is \( \frac{4 - 2}{3 - 0}=\frac{2}{3} \), y - intercept is 2. So the slope - intercept form is \( y=\frac{2}{3}x+2 \). Wait, but let's check another way. The slope - intercept form is \( y=mx + b \), \( b = 2 \). Let's take a point on the line, say \( (x=-3,y = 0) \). Plug into \( y=mx + 2 \): \( 0=m(-3)+2 \), so \( - 3m=-2 \), so \( m=\frac{2}{3} \). Yes, that's correct.

Answer:

\( y=\frac{2}{3}x + 2 \)