/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 172 Find the intercepts for each equ... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 172

Find the intercepts for each equation. \(y=\frac{1}{4} x-1\)

Short Answer

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

Step by step solution

01

Find the y-intercept

To find the y-intercept, set the value of x to 0 in the equation and solve for y. The equation is: \[ y = \frac{1}{4} x - 1 \]Substitute x = 0:\[ y = \frac{1}{4} (0) - 1 \]Therefore,\[ y = -1 \]So, the y-intercept is at the point (0, -1).
02

Find the x-intercept

To find the x-intercept, set the value of y to 0 in the equation and solve for x. The equation is: \[ y = \frac{1}{4} x - 1 \]Substitute y = 0:\[ 0 = \frac{1}{4} x - 1 \]Solve for x by isolating the variable:\[ \frac{1}{4} x = 1 \]Multiply both sides by 4:\[ x = 4 \]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.

linear equations
Linear equations are mathematical expressions that form straight lines when graphed on a coordinate plane. These equations usually have the form \ y = mx + b \ where \(m\) is the slope and \(b\) is the y-intercept.
Linear equations are fundamental in algebra as they provide a relationship between the two variables, x and y.
When we talk about the equation representing a line, it鈥檚 essentially a formula to find the y value for any given x value. One example of a linear equation is \( y = \frac{1}{4} x - 1 \). In this equation, the slope \(m\) is \ \frac{1}{4} \ and the y-intercept \(b\) is -1.
y-intercept
The y-intercept is the point where the line crosses the y-axis.
It occurs when the value of x is 0. To find the y-intercept, you simply substitute \(x = 0\) into the linear equation.
In our example \(y = \frac{1}{4} x - 1\), set \(x = 0\) which gives: \ y = \frac{1}{4} (0) - 1 \ Therefore, y is -1 when x is 0. This means the y-intercept is at point (0, -1).
The y-intercept is a crucial concept to understand as it serves as the starting point of the line on the graph when x is zero.
x-intercept
The x-intercept is the point where the line crosses the x-axis.
This happens when the value of y is 0. To find the x-intercept, you need to set \(y = 0\) and solve for \(x\) in the linear equation.
Given the equation \(y = \frac{1}{4} x - 1\), set \(y = 0\) to find:
\ 0 = \frac{1}{4} x - 1 \ Solving this equation will give you \(x = 4\).
This means the x-intercept is at point (4, 0). Identifying the x-intercept is also very important as it tells you where the line hits the x-axis when y is zero.
solving equations
Solving equations involves finding the value of the unknown variable that makes the equation true.
When dealing with linear equations, you often solve for one variable in terms of the other.
For example, to find the x-intercept in the equation \(y = \frac{1}{4} x - 1\), we set y to 0 and solve for x:
\ 0 = \frac{1}{4} x - 1 \ \ \frac{1}{4} x = 1 \ \ x = 4 \
Notice how you isolate the variable x by performing inverse operations. Conversely, finding the y-intercept means setting \(x = 0\) and solving for y:
\ y = \frac{1}{4} (0) - 1 \ \ y = -1 \
Solving equations is a critical skill in algebra as it allows you to find the exact points of intersection on a graph.

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