Problem 4 Solve each system graphically. ... [FREE SOLUTION]

Chapter 15: Problem 4

Solve each system graphically. $$ \begin{aligned} &y=2 x-5\\\ &x+y=4 \end{aligned} $$

Short Answer

Expert verified

The solution is the point where the lines intersect: (3, 1).

Step by step solution

Rewrite the equations in slope-intercept form

Rewrite both equations in the form of $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.The first equation $y = 2x - 5$ is already in slope-intercept form.The second equation $x + y = 4$ can be rewritten as: \ y = -x + 4 \.

Plot the equations on a graph

Graph the first equation $y = 2x - 5$ by plotting the y-intercept (0, -5), and using the slope 2 (rise over run) to find another point.Graph the second equation $y = -x + 4$ by plotting the y-intercept (0, 4), and using the slope -1 (rise over run) to find another point.

Identify the point of intersection

The solution to the system of equations is the point where the two lines intersect on the graph. Find and note this point of intersection.

Verify the solution

To verify, substitute the coordinates of the intersection point back into the original equations to ensure they are satisfied.

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Key Concepts

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

Slope-Intercept Form

Understanding the slope-intercept form is crucial for graphing equations. This form is written as: $y = mx + b$, where: $m$ is the slope, showing the steepness of the line, and $b$ is the y-intercept, where the line crosses the y-axis.
For example, in the equation $y = 2x - 5$, $2$ is the slope and $-5$ is the y-intercept.
Knowing the slope and y-intercept allows you to quickly plot the line on a graph.

Point of Intersection

The point of intersection is where two lines on a graph cross each other and is the solution to the system of equations.
This point satisfies both equations simultaneously.
To find it, graph each equation and look for the coordinates (x, y) where they meet.
For instance, if $y = 2x - 5$ and $y = -x + 4$ intersect at $(3, 1)$, then $x = 3$ and $y = 1$ is the solution.

Plotting on a Graph

To plot an equation on a graph, start at the y-intercept (\b b\b). For the line $y = 2x - 5$, begin at $(0, -5)$.
Next, use the slope to find another point. The slope $2$ means go up 2 units and right 1 unit from the y-intercept.
Repeat these steps to draw more points, and then, draw a straight line through them.
Do the same for the second equation.
For $y = -x + 4$, start at $(0, 4)$ and, using the slope $-1$, go down 1 unit and right 1 unit.

Solving Linear Systems

Solving linear systems can be done graphically by finding where two lines intersect.
Follow these steps:

Express each equation in slope-intercept form, if needed.
Plot each line using the y-intercept and slope.
Find the exact point where the lines cross.
Check the intersection point by plugging the coordinates back into both original equations.

This method visually shows you the solution and helps you understand the relationship between the lines.

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91影视

Solve each system graphically. $$ \begin{aligned} &y=2 x-5\\\ &x+y=4 \end{aligned} $$

Short Answer

Step by step solution

Rewrite the equations in slope-intercept form

Plot the equations on a graph

Identify the point of intersection

Verify the solution

Key Concepts

Slope-Intercept Form

Point of Intersection

Plotting on a Graph

Solving Linear Systems

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