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Chapter 5: Problem 22

Find the \(x\) - and \(y\) -intercepts of the equation. $$3 x-8=4 y$$

Short Answer

Expert verified
The y-intercept is (0, -2) and the x-intercept is ( \frac{8}{3}, 0 ).

Step by step solution

01

Find the y-intercept

To find the y-intercept, set the value of x to 0. Substitute 0 for x in the equation \[ 3x - 8 = 4y \] and solve for y.
02

Solve for y (y-intercept calculation)

Substitute x = 0 into the equation: \[ 3(0) - 8 = 4y \] which simplifies to \[ -8 = 4y \]. Divide both sides by 4 to find y: \[ y = -2 \]. So, the y-intercept is (0, -2).
03

Find the x-intercept

To find the x-intercept, set the value of y to 0. Substitute 0 for y in the equation \[ 3x - 8 = 4y \] and solve for x.
04

Solve for x (x-intercept calculation)

Substitute y = 0 into the equation: \[ 3x - 8 = 4(0) \] which simplifies to \[ 3x - 8 = 0 \]. Add 8 to both sides: \[ 3x = 8 \]. Divide both sides by 3 to find x: \[ x = \frac{8}{3} \]. So, the x-intercept is \( \frac{8}{3}, 0 \).

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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 where the graph crosses the x-axis. At this point, the value of y is always 0. To find the x-intercept for the given equation, follow these steps:
  • Set y to 0 in the equation.
  • Solve for x.
For the equation \[3x - 8 = 4y\], substitute y = 0:
\[3x - 8 = 4(0)\]This simplifies down to:
\[3x - 8 = 0\]Now, solve for x:
\[3x - 8 + 8 = 0 + 8\] which gives us:
\[3x = 8\]Divide both sides by 3:
\[x = \frac{8}{3}\]So, the x-intercept is at \(\left(\frac{8}{3}, 0\right)\).
y-intercept
To find the y-intercept of a linear equation, we look at where the graph crosses the y-axis. Here, the value of x is always 0. Here鈥檚 how to determine the y-intercept for the equation:
  • Set x to 0 in the equation.
  • Solve for y.
For the equation \[3x - 8 = 4y\], substitute x = 0:
\[3(0) - 8 = 4y\]This boils down to:
\[-8 = 4y\]Now, solve for y by dividing both sides by 4:
\[y = -2\]Therefore, the y-intercept is at (0, -2).
solving linear equations
Solving linear equations is a fundamental skill in algebra. It involves finding the values of x and y that satisfy the equation. Here鈥檚 a simple method to approach these problems:
  • Identify what you are solving for: x or y.
  • Isolate the variable on one side of the equation.
  • Simplify the equation by performing arithmetic operations.
For example, let鈥檚 use the equation from the exercise \[3x - 8 = 4y\]:
To find x given y = 0:
\[3x - 8 = 0\]Add 8 to both sides:
\[3x = 8\]Divide by 3:
\[x = \frac{8}{3}\]To find y given x = 0:
\[-8 = 4y\]Divide by 4:
\[y = -2\]
By following these steps, you can solve most linear equations effectively.

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Most popular questions from this chapter

Nature experts tell us that crickets can act as an outdoor thermometer because the rate at which a cricket chirps is linearly related to the temperature. At \(60^{\circ} \mathrm{F}\) crickets make an average of 80 chirps per minute, and at \(68^{\circ} \mathrm{F}\) they make an average of 112 chirps per minute. (a) Write an equation relating the average number of chirps \(c\) and the temperature \(t\) (b) What would be the average number of chirps at \(90^{\circ} \mathrm{F}\) ? (c) If you hear an average of 88 chirps per minute, what is the temperature? (d) Sketch a graph of this equation using the horizontal axis for \(t\) and the vertical axis for \(c\) (e) What is the \(t\) -intercept of this graph? What is the significance of this \(t\) -intercept?

Determine the slope of the line from its equation. $$y=2 x-11$$

Determine the slope of the line from its equation. $$x+y=7$$

Sketch the graph of the line satisfying the given conditions. Passing through \((2,1)\) with slope \(\frac{2}{3}\)

Write an equation of the line that passes through the point \((2,-1)\) and is parallel to the line whose equation is \(4 x-3 y=6\)
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