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Chapter 2: Problem 35

Determine the slope and the \(y\) -intercept. $$ 2 x-y=3 $$

Short Answer

Expert verified
Slope: 2, y-intercept: -3

Step by step solution

01

Write the equation in slope-intercept form

The slope-intercept form of an equation is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Given the equation: \( 2x - y = 3 \).
02

Solve for y

Isolate \( y \) on one side of the equation. Starting with \( 2x - y = 3 \), subtract \( 2x \) from both sides to get \( -y = -2x + 3 \).
03

Simplify the equation

Multiply by -1 to get \( y = 2x - 3 \). Now the equation is in the form \( y = mx + b \).
04

Identify the slope and y-intercept

In the equation \( y = 2x - 3 \), the slope \( m \) is \( 2 \), and the y-intercept \( b \) is \( -3 \).

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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 fundamental in algebra, and involve variables raised to the power of one. A linear equation typically looks like this: \text{ax + by = c}\, where \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are variables. These equations graph as straight lines. Linear equations are crucial because they help us understand relationships between quantities. For example, in the exercise given, we start with the linear equation \text{2x - y = 3}.

The goal is often to express these equations in slope-intercept form, making it easier to identify properties like the slope and \(y\)-intercept. To do this, we'll reformat the equation step-by-step.

slope
The slope of a line tells us how steep it is. It is usually represented by \(m\) in the slope-intercept form \text{y = mx + b}. Slope is calculated as the ratio of the vertical change to the horizontal change between two points on the line.

For our example, once we rearrange the given equation \text{2x - y = 3}\

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Most popular questions from this chapter

Find the domain of the function given by each equation. $$ f(x)=\frac{4 x-3}{5} $$

Determine whether each equation is linear. Find the slope of any nonvertical lines. $$ 3 x+5 y+15=0 $$

Explain why the domain of the function given by \(f(x)=\frac{x+3}{2}\) is \(\mathbb{R},\) but the domain of the function given by \(g(x)=\) \(\frac{2}{x+3}\) is not \(\mathbb{R}\)

Solve. If appropriate, classify the equation as either \(a\) contradiction or an identity. $$ 4(t-6)=7 t-15-3 t $$

Find the indicated function values. $$ F(x)=\left\\{\begin{array}{ll}{2 x,} & {\text { if } x<3} \\ {-5 x,} & {\text { if } x \geq 3}\end{array}\right. $$ a) \(F(-1) \quad\) b) \(F(3)\) c) \(F(10)\)
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