Problem 70 - Algebra for College Students

Question

Find all intercepts for each line. Some of these lines have only one intercept. $$9 x+8 y=72$$

Step-by-Step Solution

Answer

x-intercept: (8, 0), y-intercept: (0, 9)

Step 1 - Identify Intercepts

To find intercepts, identify the places where the line crosses the x-axis and y-axis. For the x-intercept, set y to 0 and solve for x. For the y-intercept, set x to 0 and solve for y.

Step 2 - Find the x-Intercept

To find the x-intercept, set y to 0 in the equation and solve for x: Rearranging gives:

Step 3 - Solve for x

Simplify the equation to solve for x: So, the x-intercept is (8, 0).

Step 4 - Find the y-Intercept

To find the y-intercept, set x to 0 in the equation and solve for y: Rearranging gives:

Step 5 - Solve for y

Simplify the equation to solve for y: So, the y-intercept is (0, 9).

Step 6 - State the Intercepts

The intercepts of the line are where it crosses the x and y axes. The x-intercept is (8, 0) and the y-intercept is (0, 9).

Key Concepts

x-intercept

To find the x-intercept of a line, you need to determine where the line crosses the x-axis. This point is found by setting y equal to 0 in the equation of the line and then solving for x.

For the given equation, \(9x + 8y = 72\), you substitute \(y = 0\) and solve for \(x\):

• Set \(y = 0\)
• Rearrange the equation to isolate \(x\): \(9x = 72\)
• Divide both sides by 9: \(x = 8\)

Thus, the x-intercept is at \((8, 0)\). This means the line crosses the x-axis at the point where \(x = 8\) and \(y = 0\).

y-intercept

The y-intercept is where the line crosses the y-axis. To find this point, you set \(x\) to 0 in the equation and solve for \(y\).

For the equation \(9x + 8y = 72\), you substitute \(x = 0\) and solve for \(y\):

• Set \(x = 0\)
• Rearrange the equation: \(8y = 72\)
• Divide both sides by 8: \(y = 9\)

Therefore, the y-intercept is at \((0, 9)\). This means the line crosses the y-axis at the point where \(x = 0\) and \(y = 9\).

solving linear equations

Solving linear equations involves finding the values of the variables that make the equation true. This can be done via various methods, but for finding intercepts, the process is straightforward.

To solve for the x-intercept and y-intercept, follow these simple steps:

• For x-intercept: Set \(y = 0\) and solve for \(x\).
• For y-intercept: Set \(x = 0\) and solve for \(y\).

The main goal is to isolate the variable we are solving for by rearranging the equation. This can involve simple arithmetic such as addition, subtraction, multiplication, or division.

graphing linear equations

Graphing a linear equation provides a visual representation of the solutions to the equation. Using intercepts makes this process more straightforward.

To graph the line \(9x + 8y = 72\), follow these steps:

• Find the x-intercept (set \(y = 0\), solve for \(x\): \((8, 0)\))
• Find the y-intercept (set \(x = 0\), solve for \(y\): \((0, 9)\))
• Plot these points on a Cartesian plane
• Draw a straight line through the points

The line through these intercepts represents all solutions to the equation. Each point \((x, y)\) on the line will satisfy the equation \(9x + 8y = 72\).