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Chapter 3: Problem 25

Find the slope and the \(y\) -intercept of each line whose equation is given. $$-3 x+y=7$$

Short Answer

Expert verified
The slope is 3 and the \(y\)-intercept is 7.

Step by step solution

01

Rewrite the Equation in Slope-Intercept Form

The slope-intercept form of a line is given by \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\)-intercept. Start by isolating \(y\) on one side of the given equation. For the equation \(-3x + y = 7\), add \(3x\) to both sides to get:\[y = 3x + 7\]
02

Identify the Slope

In the slope-intercept form \(y = mx + b\), the coefficient of \(x\) represents the slope. In the equation \(y = 3x + 7\), the slope \(m\) is \(3\).
03

Identify the \(y\)-Intercept

In the slope-intercept form \(y = mx + b\), the constant term represents the \(y\)-intercept. In the equation \(y = 3x + 7\), the \(y\)-intercept \(b\) is \(7\).

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Key Concepts

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

Slope
In mathematics, the slope is a measure of how steep a line is. Think of it as the 'tilt' of the line.
It's represented by the letter 'm' in the slope-intercept form of a line equation: \(y = mx + b\).
The slope tells us how much the y-value changes for a change in the x-value, which is often called 'rise over run'.
For example, in the equation \(y = 3x + 7\), the slope is 3, which means for every 1 unit increase in x, the y-value increases by 3 units.
Slope is crucial for understanding the direction and steepness of a line in graphing.
Y-Intercept
The y-intercept is the point where the line crosses the y-axis. It's represented by the letter 'b' in the slope-intercept form of a line equation: \(y = mx + b\).
The y-intercept gives us a starting point on the y-axis where the x-coordinate is zero.
For example, in the equation \(y = 3x + 7\), the y-intercept is 7, which means the line crosses the y-axis at the point (0, 7).
This point is important for graphing the equation and provides a reference for how the line behaves as x changes.
Linear Equations
A linear equation is an equation that creates a straight line when graphed.
It can often be written in the form \(y = mx + b\), where 'm' is the slope and 'b' is the y-intercept.
Linear equations may initially come in different forms but can be transformed into this standard form for easier interpretation.
For example, the equation \(-3x + y = 7\) can be rewritten as \(y = 3x + 7\), which is in slope-intercept form.
Understanding how to manipulate and solve linear equations is a fundamental skill in algebra.
Algebra
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols.
It allows us to solve equations and understand relationships between variables.
In the context of linear equations like \(y = 3x + 7\), algebra helps us find the slope and y-intercept.
By isolating variables and rearranging equations, we can interpret and graph linear relationships.
Mastering algebraic techniques is essential for solving various mathematical problems and understanding more complex concepts later on.

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