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Chapter 11: Problem 148

In the following exercises, find the intercepts. $$ y=\frac{1}{4} x-1 $$

Short Answer

Expert verified
The y-intercept is \((0, -1)\) and the x-intercept is \((4, 0)\).

Step by step solution

01

- Find the y-intercept

To find the y-intercept, set \(x\) to 0 and solve for \(y\). This is because the y-intercept occurs where the graph crosses the y-axis (where \(x=0\)).\[y = \frac{1}{4}(0) - 1 = -1\]So, the y-intercept is at the point \((0, -1)\).
02

- Find the x-intercept

To find the x-intercept, set \(y\) to 0 and solve for \(x\). This is because the x-intercept occurs where the graph crosses the x-axis (where \(y=0\)).\[0 = \frac{1}{4}x - 1\]Add 1 to both sides:\[1 = \frac{1}{4}x\]Multiply both sides by 4 to isolate \(x\):\[4 = x\]So, the x-intercept is at the point \((4, 0)\).

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Key Concepts

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

y-intercept
When dealing with linear equations, finding the y-intercept is one of the fundamental steps. The y-intercept is the point where the line crosses the y-axis. This happens when the value of x is zero because the y-axis is a vertical line where all x values are zero.

To find the y-intercept, simply set x to 0 in the equation and solve for y. For example, if the given equation is \( y = \frac{1}{4} x - 1 \), you set x to 0 and find y like this:

  • Replace x with 0: \( y = \frac{1}{4}(0) - 1 \)
  • Simplify: \( y = -1 \)
So, the y-intercept is at the point (0, -1). This means the line crosses the y-axis at -1.
Understanding the y-intercept helps in understanding where the line starts on the y-axis.
x-intercept
The x-intercept is another crucial point. This is where the line crosses the x-axis. The x-intercept occurs when y is zero because the x-axis is a horizontal line where all y values are zero.

To find the x-intercept, set y to 0 and solve for x. Using the same linear equation \( y = \frac{1}{4} x - 1 \), set y to 0 and solve for x:

  • Replace y with 0: \( 0 = \frac{1}{4} x - 1 \)
  • Add 1 to both sides: \( 1 = \frac{1}{4} x \)
  • Multiply both sides by 4 to isolate x: \( 4 = x \)
Therefore, the x-intercept is at the point (4, 0). This means the line crosses the x-axis at 4.
Knowing how to find the x-intercept is essential when sketching the graph of a line.
solving linear equations
Solving linear equations involves finding the values of the variables that make the equation true. Linear equations are equations of the first degree, meaning they have variables raised only to the power of one.

Here's a quick guide on how to solve them:
  • Isolate the variable: Use addition, subtraction, multiplication, or division to get the variable on one side of the equation.
  • Simplify: Combine like terms and simplify both sides of the equation as needed.
  • Check your solution: Substitute your solution back into the original equation to ensure it makes the equation true.
For instance, in the equation \( y = \frac{1}{4} x - 1 \), you may need to find specific points or intercepts as shown above. Solving linear equations is a fundamental skill in algebra that helps you understand more complex concepts later.

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Most popular questions from this chapter

In the following exercises, find the slope of each line. $$x=-4$$

Plot each point on a coordinate grid and identify the quadrant in which the point is located. a) \((3,-2)\) b) \((-3,2)\) c) \((-3,-2)\) d) \((3,2)\)

Plot each point on a coordinate grid. \((3,2)\)

In the following exercises, graph the vertical and horizontal lines. $$ y=3 $$

In the following exercises, graph the line given a point and the slope. $$ (-4,2) ; m=4 $$
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