Problem 16 - Understanding Elementary Algebra with Geometry

Question

Find the \(x\) - and \(y\) -intercepts of the equation. $$y=-3 x$$

Step-by-Step Solution

Answer

Both the x-intercept and y-intercept are (0, 0).

Step 1: Find the y-intercept

To find the y-intercept, set x to 0 and solve for y in the equation. Substituting x = 0 into the equation, we have: y = -3(0) = 0Therefore, the y-intercept is (0, 0).

Step 2: Find the x-intercept

To find the x-intercept, set y to 0 and solve for x in the equation. Substituting y = 0 into the equation, we have: 0 = -3x Solving for x, we get: x = 0Therefore, the x-intercept is (0, 0).

Key Concepts

x-intercept

The **x-intercept** of a graph is the point where the graph crosses the x-axis. At this point, the value of y is always zero. To find the x-intercept for any equation, you set y to 0 and solve for x.

For the equation given in the exercise, \( y = -3x \), you set y to 0 and solve for x:

\[ 0 = -3x \]

By dividing both sides of the equation by -3, you get:

\[ x = 0 \]

So, the x-intercept for this equation is (0, 0). This means the graph crosses the x-axis at the origin.

y-intercept

The **y-intercept** of a graph is the point where it crosses the y-axis. At this point, the value of x is zero. To find the y-intercept, you set x to 0 and solve for y.

In the exercise, the equation given is \( y = -3x \). Setting x to 0, the equation simplifies to:

\[ y = -3(0) = 0 \]

This shows that the y-intercept of the equation is (0, 0). This is where the line crosses the y-axis, which is also at the origin.

When you find that both the x-intercept and y-intercept are at the same point, (0,0) in this case, it means the line crosses the origin.

graphing

Graphing linear equations helps visualize their solutions. For the equation \ ( y = -3x ) \, we can graph using intercepts.

To start graphing, you need the intercepts:

• x-intercept: (0,0)
• y-intercept: (0,0)

Since both intercepts are the same, we need another point to draw the line. Let's use another value of x to find y.
If x = 1:

\[ y = -3(1) = -3 \]

So another point on the graph is (1, -3).

Plot the intercept (0,0) and the point (1,-3) on a graph.
Draw a straight line through these points.
This is the graph of \( y = -3x \). The line slopes downward from left to right, indicating a negative slope.