Problem 15 - Elementary Algebra

Question

Graph the equations. $$ y=-\frac{4}{3} x $$

Step-by-Step Solution

Answer

Question: Graph the linear equation $$y = -\frac{4}{3}x$$ and find its x-intercept and y-intercept. Answer: The graph of the linear equation $$y = -\frac{4}{3}x$$ has an x-intercept and y-intercept both at the origin (0, 0). The line passes through the points (0, 0) and (3, -4).

Step 1: Find the x-intercept

Set y to 0 and solve for x: $$ 0 = -\frac{4}{3}x $$ Multiply both sides by 3: $$ 0 = -4x $$ Divide both sides by -4: $$ x = 0 $$ So the x-intercept is at (0, 0).

Step 2: Find the y-intercept

Set x to 0 and solve for y: $$ y = -\frac{4}{3}(0) $$ Multiply -4/3 by 0: $$ y = 0 $$ So the y-intercept is also at (0, 0).

Step 3: Plot the intercepts and draw the line

Since both the x-intercept and y-intercept are at the origin (0, 0), we can plot this point on the graph. To get another point on the line, we can choose any value for x and plug it back into the equation to find the corresponding value of y. Let's choose x = 3: $$ y = -\frac{4}{3}(3) $$ Multiply -4/3 by 3: $$ y = -4 $$ So when x is 3, y is -4. We plot the point (3, -4) on the graph. Now, connect the points (0, 0) and (3, -4) to draw the line that represents the equation $$y = -\frac{4}{3}x$$.

Key Concepts

X-Intercept

The x-intercept of a graph is the point where the line crosses the x-axis. To find it, we set the y-coordinate to zero and solve for x. In our example with the equation
\( y = -\frac{4}{3}x \),
when we set
\( y = 0 \),
the solution reveals that
\( x = 0 \
. Therefore, the x-intercept is at the origin,
\( (0, 0) \)
. This point is crucial as it helps to start sketching the graph of the line on a Cartesian plane.

Y-Intercept

Similarly, the y-intercept is where the graph crosses the y-axis. This time, we set the x-coordinate to zero and solve for y. Returning to our equation, making x equal zero and calculating reveals that
\( y = 0 \)
as well. Hence, the y-intercept also lies at the origin, which is quite unique since lines typically have different x and y-intercepts. The y-intercept is commonly used as a starting point for graphing linear equations when plotting the line on paper or digitally.

Slope-Intercept Form

The slope-intercept form is a straightforward way of writing linear equations. It has the format
\( y = mx + b \)
, where m stands for the slope and b indicates the y-intercept. Our equation,
\( y = -\frac{4}{3}x \)
, is already in slope-intercept form with the slope
\( m = -\frac{4}{3} \)
and the y-intercept is zero since there is no b value. This form is particularly handy because it gives immediate visual cues about the graph, showing both the steepness of the line (slope) and where it hits the y-axis (y-intercept).

Plotting Points

When plotting points to draw the graph of a linear equation, start with the intercepts. After plotting the intercepts, choose another x-value, calculate the corresponding y-value, and plot this point. For our equation
\( y = -\frac{4}{3}x \):
if we let
\( x = 3 \),
then
\( y = -4 \),
so you would plot the point
\( (3, -4) \)
. Since a line is determined by two points, we can now connect the dots—with a ruler for accuracy—to show the line. In case your intercepts are the same, like in our exercise, it's especially important to plot another point to define the line's direction.