Solve This System Of Equations By Graphing. First, Graph The Equations, And Then Type The Solution.${ \begin{array}{l} y = -x + 3 \ y = \frac{5}{3}x - 5 \end{array} }$

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Introduction

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Graphing is a powerful tool for solving systems of linear equations. By graphing the equations on the same coordinate plane, we can visually identify the point of intersection, which represents the solution to the system. In this article, we will learn how to solve a system of linear equations using graphing.

Understanding the Equations

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The given system of equations consists of two linear equations:

1. $y = -x + 3$
2. $y = \frac{5}{3}x - 5$

To graph these equations, we need to understand their slopes and y-intercepts. The slope-intercept form of a linear equation is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Equation 1: $y = -x + 3$

• Slope ($m$): $-1$
• Y-intercept ($b$): $3$

Equation 2: $y = \frac{5}{3}x - 5$

• Slope ($m$): $\frac{5}{3}$
• Y-intercept ($b$): $-5$

Graphing the Equations

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To graph the equations, we can use the slope-intercept form and plot two points on each line. The first point is the y-intercept, and the second point is a point on the line that is one unit to the right of the y-intercept.

Graphing Equation 1: $y = -x + 3$

• Plot the y-intercept: $(0, 3)$
• Plot a point one unit to the right: $(1, 0)$

Draw a line through these two points to graph the equation.

Graphing Equation 2: $y = \frac{5}{3}x - 5$

• Plot the y-intercept: $(0, -5)$
• Plot a point one unit to the right: $(1, \frac{0}{3})$

Draw a line through these two points to graph the equation.

Finding the Point of Intersection

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The point of intersection represents the solution to the system of equations. To find the point of intersection, we need to find the x-coordinate and the y-coordinate of the point where the two lines intersect.

Finding the x-coordinate

To find the x-coordinate, we can set the two equations equal to each other and solve for x.

$-x + 3 = \frac{5}{3}x - 5$

Multiply both sides by 3 to eliminate the fraction:

$-3x + 9 = 5x - 15$

Add 3x to both sides:

$9 = 8x - 15$

Add 15 to both sides:

$24 = 8x$

Divide both sides by 8:

$x = 3$

Finding the y-coordinate

Now that we have the x-coordinate, we can substitute it into one of the original equations to find the y-coordinate.

Using Equation 1: $y = -x + 3$

$y = -3 + 3$

$y = 0$

Conclusion

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The point of intersection is $(3, 0)$, which represents the solution to the system of equations. By graphing the equations and finding the point of intersection, we have solved the system of linear equations.

Example Problems

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Problem 1

Solve the system of equations using graphing:

$y = 2x - 1$

$y = -x + 4$

Solution

Graph the equations and find the point of intersection. The solution is $(3, 2)$.

Problem 2

Solve the system of equations using graphing:

$y = x + 2$

$y = -2x + 1$

Solution

Graph the equations and find the point of intersection. The solution is $(-1, 3)$.

Tips and Tricks

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• When graphing the equations, make sure to plot the y-intercept and a point one unit to the right.
• When finding the point of intersection, make sure to set the two equations equal to each other and solve for x.
• When substituting the x-coordinate into one of the original equations, make sure to use the correct equation.

Conclusion

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Graphing is a powerful tool for solving systems of linear equations. By graphing the equations and finding the point of intersection, we can visually identify the solution to the system. In this article, we learned how to solve a system of linear equations using graphing and provided example problems to practice the skill.

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Q: What is the first step in solving a system of equations by graphing?

A: The first step in solving a system of equations by graphing is to graph the two equations on the same coordinate plane.

Q: How do I graph a linear equation?

A: To graph a linear equation, you need to find the slope and y-intercept of the equation. The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. Plot the y-intercept and a point one unit to the right to draw the line.

Q: What is the point of intersection?

A: The point of intersection is the point where the two lines intersect. This point represents the solution to the system of equations.

Q: How do I find the point of intersection?

A: To find the point of intersection, you need to set the two equations equal to each other and solve for x. Then, substitute the x-coordinate into one of the original equations to find the y-coordinate.

Q: What if the lines are parallel?

A: If the lines are parallel, they will never intersect, and there will be no solution to the system of equations.

Q: What if the lines intersect at a point that is not an integer?

A: If the lines intersect at a point that is not an integer, you can use a calculator or a graphing tool to find the exact coordinates of the point of intersection.

Q: Can I use graphing to solve systems of equations with more than two variables?

A: No, graphing is only used to solve systems of linear equations with two variables.

Q: What are some common mistakes to avoid when graphing systems of equations?

A: Some common mistakes to avoid when graphing systems of equations include:

• Graphing the equations incorrectly
• Failing to find the point of intersection
• Substituting the x-coordinate into the wrong equation
• Not checking for parallel lines

Q: How can I check my work when graphing systems of equations?

A: You can check your work by:

• Graphing the equations again to make sure they are correct
• Using a calculator or a graphing tool to find the point of intersection
• Substituting the x-coordinate into one of the original equations to find the y-coordinate

Q: What are some real-world applications of solving systems of equations by graphing?

A: Some real-world applications of solving systems of equations by graphing include:

• Finding the intersection of two roads or paths
• Determining the point of intersection of two lines of sight
• Solving problems in physics, engineering, and economics

Q: Can I use graphing to solve systems of equations with non-linear equations?

A: No, graphing is only used to solve systems of linear equations. Non-linear equations require different methods, such as substitution or elimination.

Q: What are some tips for graphing systems of equations?

A: Some tips for graphing systems of equations include:

• Use a ruler or a straightedge to draw the lines
• Make sure to plot the y-intercept and a point one unit to the right
• Use a calculator or a graphing tool to find the point of intersection
• Check your work carefully to avoid mistakes

Q: Can I use graphing to solve systems of equations with fractions?

A: Yes, you can use graphing to solve systems of equations with fractions. Simply multiply both sides of the equation by the least common multiple of the denominators to eliminate the fractions.

Q: What are some common mistakes to avoid when graphing systems of equations with fractions?

A: Some common mistakes to avoid when graphing systems of equations with fractions include:

• Failing to multiply both sides of the equation by the least common multiple of the denominators
• Graphing the equations incorrectly
• Failing to find the point of intersection

Q: How can I check my work when graphing systems of equations with fractions?

A: You can check your work by:

• Graphing the equations again to make sure they are correct
• Using a calculator or a graphing tool to find the point of intersection
• Substituting the x-coordinate into one of the original equations to find the y-coordinate