Problem 72 - Algebra for College Students

Question

Find all intercepts for each line. Some of these lines have only one intercept. $$y=-4 x$$

Step-by-Step Solution

Answer

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

Step 1: Understand the equation

The given equation of the line is: \( y = -4x \). This is in the slope-intercept form \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept.

Step 2: Find the y-intercept

To find the y-intercept, set \(x = 0\) and solve for \(y\). \( y = -4(0) = 0 \). So the y-intercept is \( (0, 0) \).

Step 3: Find the x-intercept

To find the x-intercept, set \(y = 0\) and solve for \(x\). \( 0 = -4x \). Solving this gives you \( x = 0 \). So the x-intercept is also \( (0, 0) \).

Step 4: Conclusion

The intercepts of the line \( y = -4x \) are both at the point \( (0, 0) \). This means the line passes through the origin.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a line is a way of writing the equation of a line. It looks like this: y = mx + bWhere *m* is the slope, and *b* is the y-intercept. The slope *m* tells you how steep the line is and the direction it goes. Positive slopes go up from left to right, and negative slopes go down. In the given equation y = -4xthe slope *m* is -4. This means for every unit you move right on the x-axis, you move 4 units down on the y-axis. The y-intercept *b* is 0 in this equation, which indicates that the line crosses the y-axis at the origin (0,0).

X-Intercept

The x-intercept is the point where the line crosses the x-axis. To find the x-intercept, set y to 0 and solve for x. From our equation y = -4xset y to 0:0 = -4xSolving for x, you divide both sides by -4:x = 0So the x-intercept is (0,0). This means that the line crosses the x-axis exactly at the origin.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis. To find the y-intercept, set x to 0 and solve for y. Using our equation y = -4xset x to 0:y = -4(0)y = 0So, the y-intercept is (0,0). This confirms what we saw with the x-intercept: the line passes through the origin. In this particular line, the origin is both the x-intercept and the y-intercept.