/*! 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 48 Write and graph a linear equatio... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 48

Write and graph a linear equation that has \((-2,4)\) as a solution.

Short Answer

Expert verified
The equation is \( y = x + 6 \). It passes through \((-2,4)\) and can be graphed starting at \((0,6)\) with a slope of 1.

Step by step solution

01

Understand the task

We need to write and graph a linear equation. The equation must be linear, meaning it can be written in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. The point \((-2,4)\) must lie on this line, which means it satisfies the equation.
02

Choose a slope

Since there are an infinite number of lines that can pass through a single point, we can choose any slope. For simplicity, let's choose the slope \( m = 1 \).
03

Find the y-intercept

Use the point \((-2,4)\) and the slope to find the y-intercept \( b \). Substitute \( x = -2 \), \( y = 4 \), and \( m = 1 \) into the equation \( y = mx + b \):\[4 = 1(-2) + b\]Solve for \( b \):\[4 = -2 + b \6 = b\]So, the y-intercept is \( b = 6 \).
04

Write the equation

Now that we have the slope \( m = 1 \) and the y-intercept \( b = 6 \), we can write the equation of the line:\[y = 1x + 6\]
05

Graph the equation

To graph the line \( y = x + 6 \), start by plotting the y-intercept, which is the point \((0,6)\). From this point, use the slope \( m = 1 \), which means go up 1 unit and to the right 1 unit, repeatedly to get more points, such as \((1,7)\), \((2,8)\), etc. Plot these points and draw a straight line through them. Ensure the point \((-2,4)\) is on this line.

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form is a fundamental concept in understanding linear equations. It provides a straightforward way to express a linear equation and offers immediate insight into the line it represents. A linear equation can be written in the form \( y = mx + b \), where:
  • \( m \) is the slope of the line.
  • \( b \) is the y-intercept, which means the point where the line crosses the y-axis.
The slope-intercept form is intuitive because it uses the slope \( m \), which tells us how steep the line is. A positive slope means the line ascends from left to right, while a negative slope means it descends. The y-intercept \( b \) reveals where the line starts on the y-axis. Choosing a relevant slope and finding the y-intercept based on known points allows you to craft any linear equation effortlessly.
Graphing Linear Equations
Graphing linear equations is an essential skill that provides a visual representation of equations. Once you have a linear equation in slope-intercept form, such as \( y = x + 6 \), you can proceed with graphing it. To graph this equation:
  • Begin at the y-intercept \((0,6)\). This is your starting point on the y-axis.
  • Using the slope \( m = 1 \), move up one unit and right one unit to plot additional points.
  • Continue this pattern to generate more points like \((1,7)\), \((2,8)\), etc.
After plotting a few points, draw a straight line through them. It is helpful to ensure alignment with your initial point, \((-2,4)\), verifying the accuracy of your equation. Graphing not only cements understanding but also demonstrates the harmony between algebraic expressions and their graphical counterparts.
Coordinate Points
Coordinate points are crucial in understanding the geometry of the graph. A coordinate point is composed of two values; \((x, y)\). It represents a specific location on the Cartesian plane:
  • The first number, \( x \), measures horizontal distance from the origin.
  • The second number, \( y \), measures vertical distance from the origin.
When working with a point like \((-2,4)\), here's how it works:
  • The \( x \) value \(-2\) indicates 2 units to the left of the origin.
  • The \( y \) value \(4\) shows 4 units above the origin.
These points act as anchors when plotting equations. They confirm the intersection of plotted lines and ensure the line truly represents the equation. By checking a point like \((-2,4)\) on the graph, you affirm the line鈥檚 correctness. They're not just numbers; they indicate the line鈥檚 specific position in space.

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Most popular questions from this chapter

Graph each equation using the slope and \(y\) -intercept. $$-2 x+y=-1$$

MUSIC Aisha has 50 dollars to spend on music. single songs cost 1 dollars to download and entire CDs cost 10 dollars Find an equation to represent the number of single songs \(x\) and the number of CDs \(y\) Aisha can buy with 50 dollars .Then, find three ordered pairs that satisfy this condition.

State the slope and the \(y\) -intercept of the graph of each equation. $$y=4$$

Write an equation in slope-intercept form for each line. slope \(=-2, y\) -intercept \(=2\)

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