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Chapter 3: Problem 74

Find the slope and the y-intercept of the graph of each equation and graph it. See Examples 4 and 5. $$ 4 x-2 y=6 $$

Short Answer

Expert verified
The slope is 2 and the y-intercept is -3.

Step by step solution

01

Convert the Equation to Slope-Intercept Form

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. We start with the equation \( 4x - 2y = 6 \). To convert this to the slope-intercept form, we must solve for \( y \). We will isolate \( y \) by moving \( 4x \) to the other side and then dividing by \(-2\). This gives us: \(-2y = -4x + 6\). Divide the entire equation by \(-2\): \[ y = 2x - 3 \].
02

Identify the Slope and Y-Intercept

Now that the equation is in the form \( y = mx + b \), we can identify the slope \( m \) and the y-intercept \( b \). From \( y = 2x - 3 \), the slope \( m \) is \( 2 \) and the y-intercept \( b \) is \(-3\).
03

Graph the Equation

To graph the equation, start by plotting the y-intercept on the y-axis. Plot the point \( (0, -3) \) since the y-intercept is \(-3\). From this point, use the slope to find another point on the line. A slope of \( 2 \) means you rise 2 units for every 1 unit you move to the right. From \( (0, -3) \), move up 2 units and 1 unit to the right to locate the point \( (1, -1) \). Draw a line through these two points to complete the graph.

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Key Concepts

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

Graphing Linear Equations
Graphing linear equations is a fundamental skill in algebra. The equation of a line gives us a direct way to sketch it on a graph. Understanding the structure of a linear equation helps to quickly identify key characteristics of the line. To graph a linear equation, such as the example given
  • Step 1: Start by converting the given equation into slope-intercept form (if needed). This form makes it easier to identify crucial graphing features.
  • Step 2: Plot the y-intercept on a coordinate plane.
  • Step 3: Use the slope to find another point on the line.
  • Step 4: Draw a line through the points to represent the equation.
By following these steps, you can effectively graph most linear equations on a standard Cartesian coordinate plane.
Identifying Slope and Y-Intercept
The slope and y-intercept are two essential components of any linear equation. They determine the line's direction, steepness, and position on the graph. The slope-intercept form, which is written as \( y = mx + b \), makes it easy to spot these components:
  • Slope \( (m) \): This is the number that represents the steepness or angle of the line. A positive slope climbs upwards as you move from left to right, while a negative slope descends.
  • Y-Intercept \( (b) \): This is where the line crosses the y-axis. It's simply the value of \( y \) when \( x = 0 \).
Identifying these parts in an equation such as \( y = 2x - 3 \) tells us the slope is 2 and the y-intercept is -3.Once you know these, you can easily plot the line and understand its characteristics.
Converting Equations to Slope-Intercept Form
Converting an equation into slope-intercept form is a valuable process that simplifies graphing and interpreting the equation's features. This form is especially useful because it allows direct identification of the slope and y-intercept. Let's break down the process using the example equation \( 4x - 2y = 6 \):
  • First, rearrange the equation to solve for \( y \). This means moving the \( x \) term to the right side so you can isolate \( y \).
  • In our example, this means rewriting the equation as \(-2y = -4x + 6\).
  • Next, divide every term by the coefficient of \( y \), which in this case is -2, leading to \( y = 2x - 3 \).
Once in this form, you can quickly identify that the slope is 2 and the y-intercept is -3, allowing for straightforward graphing and analysis of the line.

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Most popular questions from this chapter

Two planes are \(6,000\) miles apart, and their speeds differ by \(200 \mathrm{mph}\). They travel toward each other and meet in 5 hours. Find the speed of the slower plane.

A sporting goods manufacturer allocates at least \(2,400\) units of production time per day to make baseballs and footballs. It takes 20 units of time to make a baseball and 30 units of time to make a football. If \(x\) represents the number of baseballs made and \(y\) represents the number of footballs made, the graph of \(20 x+30 y \geq 2,400\) shows the possible ways to schedule the production time. Graph the inequality. Then find three possible combinations of production time for the company to make baseballs and footballs.

Vacationing. The function \(C(d)=500+100(d-3)\) gives the cost in dollars to rent an RV motor home for \(d\) days. (The terms of the rental agreement require a 3 -day minimum.) Find the cost of renting the RV for a vacation that will last 7 days. CAN'T COPY THE IMAGE

On a quiz, a student was asked to find the slope of the graph of \(y=2 x+3 .\) She answered: \(m=2 x .\) Her instructor marked it wrong. Explain why the answer is incorrect.

Find the slope of a line perpendicular to the line passing through the given two points. See Example \(9 .\) \(\left(-2, \frac{1}{2}\right)\) and \(\left(-1, \frac{3}{2}\right)\)
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