Questions: a. Graph the line that passes through the points (4,2) and (3,3). b. What are the x - and y-intercept of this line?

Transcript text: a. Graph the line that passes through the points $(4,2)$ and $(3,3)$. b. What are the $x$ - and $y$-intercept of this line?

Solution

Solution Steps

Step 1: Identify the given points

The given points are (4, 2) and (3, 3).

Step 2: Calculate the slope of the line

The slope $ m $ is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the given points: \[ m = \frac{3 - 2}{3 - 4} = \frac{1}{-1} = -1 \]

Step 3: Use the point-slope form to find the equation of the line

The point-slope form of the equation of a line is: \[ y - y_1 = m(x - x_1) \] Using point (4, 2) and slope $ m = -1 $: \[ y - 2 = -1(x - 4) \] \[ y - 2 = -x + 4 \] \[ y = -x + 6 \]

Step 4: Graph the line

Plot the points (4, 2) and (3, 3) on the graph. Draw a line through these points. The line should have a slope of -1, meaning it goes down 1 unit for every 1 unit it goes to the right.

Step 5: Find the x-intercept and y-intercept

Y-intercept: Set $ x = 0 $ in the equation $ y = -x + 6 $: \[ y = -0 + 6 = 6 \] So, the y-intercept is (0, 6).
X-intercept: Set $ y = 0 $ in the equation $ y = -x + 6 $: \[ 0 = -x + 6 \] \[ x = 6 \] So, the x-intercept is (6, 0).