Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the \(x\) -intercept and \(y\) -intercept of the graph of \(5 x+7 y=140\).

Short answer

x-intercept: (28, 0), y-intercept: (0, 20)

Step-by-step solution

Understand the intercepts

The intercepts are points where the graph crosses the axes. For the x-intercept, the value of y is 0. For the y-intercept, the value of x is 0.

Find the x-intercept

To find the x-intercept, set y = 0 in the equation and solve for x:\(5x + 0 = 140\)\(5x = 140\)\(x = \frac{140}{5}\)\(x = 28\).So, the x-intercept is (28, 0).

Find the y-intercept

To find the y-intercept, set x = 0 in the equation and solve for y:\(0 + 7y = 140\)\(7y = 140\)\(y = \frac{140}{7}\)\(y = 20\).So, the y-intercept is (0, 20).

Summarize the intercepts

The x-intercept is at (28, 0) and the y-intercept is at (0, 20).

Key concepts

x-intercept

The x-intercept is where the graph of the equation crosses the x-axis. At this point, the value of y is always 0. To find the x-intercept of the equation, you simply substitute y with 0 and then solve the resulting equation for x. Here’s how it works with our given linear equation, which is 5x + 7y = 140.

First, set y to 0:
5x + 7(0) = 140
5x = 140.
Now, solve for x by dividing both sides by 5:
x = 140 / 5
x = 28.
Thus, the x-intercept is at the point (28, 0).

Understanding how to find the x-intercept allows us to determine one of the key points where our graph crosses the horizontal axis.

y-intercept

The y-intercept is where the graph of the equation crosses the y-axis. At the y-intercept, the value of x is always 0. To find the y-intercept of an equation, substitute x with 0 and then solve for y. Applying this to our given linear equation 5x + 7y = 140, you would:

Set x to 0:
5(0) + 7y = 140
7y = 140.
Solve for y by dividing both sides by 7:
y = 140 / 7
y = 20.
Therefore, the y-intercept is at the point (0, 20).

Knowing how to find the y-intercept is crucial for understanding where the graph crosses the vertical axis. This point provides valuable insight into the behavior of linear equations.

solving linear equations

Solving linear equations involves finding the values of the variables that make the equation true. For a linear equation in two variables like 5x + 7y = 140, you can find particular solutions by isolating each variable.

When solving for the x-intercept, as we see from above, you substitute y = 0 and solve for x. This gives us a specific point on the graph. Similarly, when solving for the y-intercept, you substitute x = 0 and solve for y.

In both cases, you are applying basic algebraic principles to isolate the variable. Here are the general steps for solving such equations:

• Identify which variable to isolate (either x or y).
• Substitute the other variable with 0 if looking for intercepts.
• Perform the algebraic operations needed to solve for the isolated variable.

Mastery of solving linear equations not only helps in plotting graphs but also builds a foundational skill for understanding more complex equations.