Solve This System Of Equations By Graphing. First, Graph The Equations, And Then Type The Solution.$\[ \begin{array}{l} y = X - 4 \\ y = -\frac{1}{6}x + 3 \end{array} \\]

Feb 28, 2025 by ADMIN 171 views

Introduction

In mathematics, a system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. There are several methods to solve a system of equations, including substitution, elimination, and graphing. In this article, we will focus on solving a system of equations by graphing. We will first graph the equations and then type the solution.

Graphing the Equations

To graph the equations, we need to find the x and y intercepts of each equation. The x-intercept is the point where the graph crosses the x-axis, and the y-intercept is the point where the graph crosses the y-axis.

Equation 1: y = x - 4

To find the x-intercept, we set y = 0 and solve for x.

0 = x - 4 x = 4

So, the x-intercept of the first equation is (4, 0).

To find the y-intercept, we set x = 0 and solve for y.

y = 0 - 4 y = -4

So, the y-intercept of the first equation is (0, -4).

Equation 2: y = -\frac{1}{6}x + 3

To find the x-intercept, we set y = 0 and solve for x.

0 = -\frac{1}{6}x + 3 \frac{1}{6}x = 3 x = 18

So, the x-intercept of the second equation is (18, 0).

To find the y-intercept, we set x = 0 and solve for y.

y = -\frac{1}{6}(0) + 3 y = 3

So, the y-intercept of the second equation is (0, 3).

Graphing the Equations

Now that we have the x and y intercepts of each equation, we can graph the equations.

Graph of Equation 1: y = x - 4

The graph of the first equation is a straight line with a slope of 1 and a y-intercept of -4.

Graph of Equation 2: y = -\frac{1}{6}x + 3

The graph of the second equation is a straight line with a slope of -\frac{1}{6} and a y-intercept of 3.

Finding the Solution

To find the solution to the system of equations, we need to find the point of intersection of the two graphs.

From the graph, we can see that the two lines intersect at the point (12, -2).

Conclusion

In this article, we solved a system of equations by graphing. We first graphed the equations and then found the point of intersection of the two graphs. The solution to the system of equations is (12, -2).

Tips and Tricks

When graphing the equations, make sure to label the x and y intercepts.
When finding the solution, make sure to check if the point of intersection is a valid solution.
When graphing the equations, make sure to use a ruler or a straightedge to draw the lines.

Real-World Applications

Solving a system of equations by graphing has many real-world applications. For example, in physics, we can use graphing to solve problems involving motion and forces. In engineering, we can use graphing to solve problems involving stress and strain.

Common Mistakes

When graphing the equations, make sure to label the x and y intercepts.
When finding the solution, make sure to check if the point of intersection is a valid solution.
When graphing the equations, make sure to use a ruler or a straightedge to draw the lines.

Conclusion

Introduction

In our previous article, we solved a system of equations by graphing. We first graphed the equations and then found the point of intersection of the two graphs. In this article, we will answer some common questions related to solving a system of equations by graphing.

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 equations. This involves finding the x and y intercepts of each equation and then drawing the lines on a coordinate plane.

Q: How do I find the x and y intercepts of an equation?

A: To find the x-intercept of an equation, set y = 0 and solve for x. To find the y-intercept of an equation, set x = 0 and solve for y.

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 a valid solution?

A: If the lines intersect at a point that is not a valid solution, then the point of intersection is not a valid solution to the system of equations.

Q: Can I use graphing to solve a system of equations with more than two equations?

A: Yes, you can use graphing to solve a system of equations with more than two equations. However, it may be more difficult to find the point of intersection of three or more lines.

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

A: Some common mistakes to avoid when graphing include:

Not labeling the x and y intercepts
Not using a ruler or straightedge to draw the lines
Not checking if the point of intersection is a valid solution

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

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

Physics: solving problems involving motion and forces
Engineering: solving problems involving stress and strain
Economics: solving problems involving supply and demand

Q: Can I use graphing to solve a system of equations with variables other than x and y?

A: Yes, you can use graphing to solve a system of equations with variables other than x and y. However, you will need to use a 3D coordinate plane and graph the equations in three dimensions.

Conclusion

In conclusion, solving a system of equations by graphing is a useful technique that has many real-world applications. By following the steps outlined in this article and avoiding common mistakes, you can solve a system of equations by graphing and find the solution.

Tips and Tricks

When graphing the equations, make sure to label the x and y intercepts.
When finding the solution, make sure to check if the point of intersection is a valid solution.
When graphing the equations, make sure to use a ruler or straightedge to draw the lines.