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Chapter 0: Problem 79

Find the \(x\) - and \(y\) -intercepts for the line \(4 x-6 y=40\).

Short Answer

Expert verified
x-intercept: (10, 0); y-intercept: (0, -\frac{20}{3})

Step by step solution

01

- Find the x-intercept

To find the x-intercept, set the value of y to 0 and solve for x. The equation is: \[4x - 6(0) = 40\]This simplifies to: \[4x = 40\]Now, divide both sides by 4: \[x = 10\]So, the x-intercept is (10, 0).
02

- Find the y-intercept

To find the y-intercept, set the value of x to 0 and solve for y. The equation is: \[4(0) - 6y = 40\]This simplifies to: \[-6y = 40\]Now, divide both sides by -6: \[y = -\frac{40}{6}\]Simplifying further: \[y = -\frac{20}{3}\]So, the y-intercept is (0, -\frac{20}{3}).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

x-intercept
The x-intercept of a linear equation is the point where the graph of the equation crosses the x-axis. To find the x-intercept, set the value of y to 0 and solve the equation for x. This is because any point on the x-axis has a y-coordinate of 0. In our example, we start with the equation:

\[4x - 6y = 40\]

By setting y to 0, the equation simplifies to:

\[4x - 6(0) = 40\]

This further simplifies to:

\[4x = 40\]

Then, by dividing both sides by 4, we get:

\[x = 10\]

Thus, the x-intercept is the point (10, 0). This indicates that the line crosses the x-axis at the point where x is 10.
y-intercept
The y-intercept of a linear equation is the point where the graph crosses the y-axis. To find the y-intercept, set the value of x to 0 and solve the equation for y. This works because any point on the y-axis has an x-coordinate of 0. In our example, we start with the equation:

\[4x - 6y = 40\]

By setting x to 0, the equation simplifies to:

\[4(0) - 6y = 40\]

Which simplifies to:

\[-6y = 40\]

Next, we divide both sides by -6 to isolate y:

\[y = -\frac{40}{6}\]

Simplifying further, we get:

\[y = -\frac{20}{3}\]

Thus, the y-intercept is the point \((0, -\frac{20}{3})\). This tells us that the line crosses the y-axis at the point where y is \(-\frac{20}{3}\).
solving linear equations
Solving linear equations often involves finding the values of x and y that satisfy the equation. The general form of a linear equation in two variables is \(Ax + By = C\), where A, B, and C are constants.

Here are key steps to solve and understand such equations:

  • Isolate the variable: Make either x or y the subject of the formula by using basic algebraic operations like addition, subtraction, multiplication, or division.
  • Intercepts: Finding the x- and y-intercepts involves setting y and x to zero, respectively, in the equation to find the points where the line crosses the axes.
  • Plotting: These intercepts can be used to graph the line on a coordinate plane, giving a visual representation of the solution.
  • Verify your solution: Finally, plugging the intercepts or other solutions back into the original equation ensures that they satisfy the equation.


In the equation \((4x - 6y = 40)\), we have demonstrated how to find both intercepts:
  • For the x-intercept: Setting y to 0 gives x = 10.
  • For the y-intercept: Setting x to 0 gives y = -\(\frac{20}{3}\).

These points help in graphing the line accurately and understanding where it cuts across the axes.

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