/*! 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 Solve by graphing. $$ \begin... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 48

Solve by graphing. $$ \begin{aligned} &x+2 y=9 \\ &y=5 x-12 \end{aligned} $$

Short Answer

Expert verified
The lines intersect at (3, 3).

Step by step solution

01

- Rewrite equations in slope-intercept form

The first equation is already in standard form. Rewrite it in slope-intercept form, which is y = mx + b.For the first equation:\[x + 2y = 9\]Solve for y:\[2y = -x + 9\]\[y = -\frac{1}{2}x + \frac{9}{2}\]The second equation is already in slope-intercept form:\[y = 5x - 12\]
02

- Graph the first equation

To graph \(y = -\frac{1}{2}x + \frac{9}{2}\), start with the y-intercept. Here, the y-intercept is \(\frac{9}{2} (or 4.5)\).Then use the slope \(-\frac{1}{2}\), which means for every 2 units you go to the right, go down 1 unit. Plot several points and draw the line.
03

- Graph the second equation

To graph \(y = 5x - 12\), start with the y-intercept, which is -12.Then use the slope 5, which means for every 1 unit you go to the right, go up 5 units. Plot several points and draw the line.
04

- Find the intersection point

Look at the graph and identify the point where both lines intersect. The coordinates of this point are the solution to the system of equations.

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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 one of the most commonly used forms to write the equation of a line. It is expressed as \(y = mx + b\).
This form is incredibly useful as it directly shows the slope (m) and the y-intercept (b) of the line.
The slope indicates how steep the line is, while the y-intercept is where the line crosses the y-axis.
To convert an equation into slope-intercept form, you must solve for y.
For example, from the exercise provided, the equation \[x + 2y = 9\] can be rewritten into slope-intercept form:
\[2y = -x + 9\]
\[y = -\frac{1}{2}x + \frac{9}{2}\].
This tells us that the slope (m) is \(-\frac{1}{2}\) and the y-intercept (b) is \(\frac{9}{2}\).
graphing linear equations
Graphing linear equations is a visual method to understand how two quantities relate to each other. When you graph an equation, you plot points on a coordinate grid.
Here's a simple process to graph a linear equation:
  • Rewrite the equation in slope-intercept form \(y = mx + b\), if it isn't already.
  • Identify the y-intercept (b), which is the point where the line crosses the y-axis.
  • Use the slope (m) to determine the rise over run; this tells you how to move from the y-intercept to plot other points.
  • Plot the y-intercept and use the slope to find at least two more points.
  • Connect these points to draw the line.
In our exercise, the first equation \(y = -\frac{1}{2}x + \frac{9}{2}\) and the second equation \(y = 5x - 12\) have been graphed in this way.
y-intercept
The y-intercept is a key component when graphing linear equations. It represents the point where the line crosses the y-axis.
In the slope-intercept form \(y = mx + b\), the variable b signifies the y-intercept.
In our given problems:
\

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