Problem 14 - Introductory Algebra for College Students

Question

Find the \(x\)-intercept and the \(y\)-intercept of the graph of each equation. Do not graph the equation. \(-x+3 y=-10\)

Step-by-Step Solution

Answer

The x-intercept for the given equation is 10 and the y-intercept is \(-10/3\).

Step 1: Finding the x-intercept

Set \(y = 0\) in the equation \(-x+3(0)=-10\). Simplifying this gives the equation to be solved for \(x\) which will give the x-intercept: \(-x = -10\). Solving for \(x\) gives us \(x = 10\). Thus, the x-intercept for the given equation is 10.

Step 2: Finding the y-intercept

Set \(x = 0\) in the equation \(-0+3 y=-10\). Simplifying this gives you the equation to be solved for \(y\) which will give the y-intercept: \(3y = -10\). Solving for \(y\) gives us \(y = -10/3\). So, the y-intercept for the given equation is \(-10/3\)

Key Concepts

x-intercept

The x-intercept is a fundamental concept in understanding linear equations. It refers to the point on a graph where the line crosses the x-axis. At this point, the value of y is always zero.

To find the x-intercept, you simply set y to zero in the equation and solve for x. It's like asking, "Where does the line meet the floor, assuming that the floor represents the x-axis?" This is because on the x-axis, the value of y is nonexistent—think of it as staying grounded on the floor without any vertical lift!

Here's how to calculate it, using the given equation as an example:

• Start with the equation: -x + 3y = -10
• Set y to 0: -x + 3(0) = -10
• Simplify and solve for x: -x = -10
  So, x = 10.

Thus, the x-intercept is at x = 10.

y-intercept

The y-intercept is another key idea for interpreting linear equations. It represents where the line cuts through the y-axis. At this particular point, the value of x is always zero.
The y-intercept is easy to determine by setting x to zero in the equation, then solving for y. This process is somewhat like finding out where a water fountain shoots water straight up to meet a vertical wall—it happens right where x is zero, acting purely on vertical motion.
To illustrate this, consider our equation:

• Begin with the equation: -x + 3y = -10
• Set x to 0: -0 + 3y = -10
• Solve the equation for y: 3y = -10
  Simplified, y = -10/3.

Thus, the y-intercept is at y = -10/3.

linear equation

A linear equation is an algebraic expression that represents a straight line when graphed on a coordinate plane. It usually comes in the form of "ax + by = c," where a, b, and c are constants. These kinds of equations are key in algebra because they form the basis for more complex mathematical concepts.
When dealing with a linear equation, each term is either a constant or the product of a constant and the first power of a variable. There are no squares, cubes, or higher powers, which is what gives the graph of a linear equation its characteristic straight-line appearance.
In practical terms:

• The coefficients (like -1 and 3 in our example equation) determine the slope and intercepts of the line.
• The solution or solutions to the equation are the coordinates of points where the line crosses the axes—specifically, the x-intercept and y-intercept.

Understanding linear equations is crucial because they model many real-world situations, like business profits, moving objects, or even shadows moving with the day!