Problem 5 Solve each system graphically. ... [FREE SOLUTION]

Chapter 15: Problem 5

Solve each system graphically. $$ \begin{aligned} &y=x-3\\\ &y=-2 x+3 \end{aligned} $$

Short Answer

Expert verified

The solution to the system is the point (2, -1).

Step by step solution

- Plot the First Equation

Begin by plotting the equation $y = x - 3$. This is a linear equation in slope-intercept form $y = mx + b$, where the slope $m$ is 1 and the y-intercept $b$ is -3. To plot this, start at the point (0, -3) on the y-axis and use the slope to find another point. Since the slope is 1, go up 1 unit and right 1 unit to get the point (1, -2). Draw a straight line through these points.

- Plot the Second Equation

Next, plot the equation $y = -2x + 3$. This is also a linear equation in slope-intercept form, where the slope $m$ is -2 and the y-intercept $b$ is 3. Start at the point (0, 3) on the y-axis. From there, use the slope to find another point. Since the slope is -2, go down 2 units and right 1 unit to get the point (1, 1). Draw a straight line through these points.

- Find the Intersection Point

To solve the system graphically, look for the point where the two lines intersect. The intersection of the lines $y = x - 3$ and $y = -2x + 3$ represents the solution to the system. By examining the graph, you can see that the lines intersect at the point (2, -1).

- Verify the Solution

To ensure the solution is correct, substitute the coordinates of the intersection point (2, -1) back into both equations. For $y = x - 3$: $-1 = 2 - 3$ which is true.For $y = -2x + 3$: $-1 = -2(2) + 3$ which is also true. Therefore, the solution is verified.

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Key Concepts

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

graphical method

The graphical method is a way to solve a system of linear equations by drawing their graphs on the same set of axes. This method involves plotting each equation as a straight line and finding the point where these lines intersect. Here鈥檚 why it鈥檚 useful: it gives you a visual representation of the solution.

To start, convert each equation into its slope-intercept form if it鈥檚 not already. This form makes it easier to plot because it directly shows the y-intercept and the slope. Once both lines are plotted, the intersection point of the lines represents the solution to the system.

The graphical method is simple, but it may not always be the most precise since plotting by hand can introduce tiny errors. However, it's a great way to understand the basic concept of solving systems of equations.

slope-intercept form

The slope-intercept form of a linear equation is one of the most common and useful forms to work with. This form is written as:

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91影视

Solve each system graphically. $$ \begin{aligned} &y=x-3\\\ &y=-2 x+3 \end{aligned} $$

Short Answer

Step by step solution

- Plot the First Equation

- Plot the Second Equation

- Find the Intersection Point

- Verify the Solution

Key Concepts

graphical method

slope-intercept form

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