Select The Correct Answer From Each Drop-down Menu.Use The System Of Equations Below To Complete The Sentence.${ \begin{aligned} x - 2y &= 5 \ 3x + 15y &= -6 \end{aligned} }$Graph AGraph BGraph CGraph DThe Graph That Correctly Represents

Mar 11, 2025 by ADMIN 239 views

**Solving Systems of Equations: A Graphical Approach**

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. One of the methods used to solve systems of equations is the graphical approach, where the equations are graphed on a coordinate plane and the point of intersection is found. In this article, we will use the graphical approach to solve a system of equations and select the correct answer from each drop-down menu.

The System of Equations

The system of equations given is:

\begin{aligned} x - 2y &= 5 \\ 3x + 15y &= -6 \end{aligned}

To solve this system of equations, we can use the graphical approach. We will graph each equation on a coordinate plane and find the point of intersection.

Graphing the Equations

Graph A

To graph the first equation, $x - 2y = 5$ , we can rewrite it in slope-intercept form, $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

y = \frac{1}{2}x - \frac{5}{2}

The slope of this line is $\frac{1}{2}$ and the y-intercept is $-\frac{5}{2}$ . We can graph this line on a coordinate plane.

Graph B

To graph the second equation, $3x + 15y = -6$ , we can rewrite it in slope-intercept form, $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

y = -\frac{1}{5}x - \frac{2}{5}

The slope of this line is $-\frac{1}{5}$ and the y-intercept is $-\frac{2}{5}$ . We can graph this line on a coordinate plane.

Graph C

To graph the third equation, $x - 2y = 5$ , we can rewrite it in slope-intercept form, $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

y = \frac{1}{2}x - \frac{5}{2}

The slope of this line is $\frac{1}{2}$ and the y-intercept is $-\frac{5}{2}$ . We can graph this line on a coordinate plane.

Graph D

To graph the fourth equation, $3x + 15y = -6$ , we can rewrite it in slope-intercept form, $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

y = -\frac{1}{5}x - \frac{2}{5}

The slope of this line is $-\frac{1}{5}$ and the y-intercept is $-\frac{2}{5}$ . We can graph this line on a coordinate plane.

Finding the Point of Intersection

To find the point of intersection, we need to find the point where the two lines intersect. We can do this by finding the x-coordinate and y-coordinate of the point of intersection.

Let's find the point of intersection of the two lines.

Finding the x-coordinate

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

\frac{1}{2}x - \frac{5}{2} = -\frac{1}{5}x - \frac{2}{5}

Solving for x, we get:

x = 1

Finding the y-coordinate

To find the y-coordinate of the point of intersection, we can substitute the x-coordinate into one of the equations and solve for y.

Substituting x = 1 into the first equation, we get:

y = \frac{1}{2}(1) - \frac{5}{2}

Solving for y, we get:

y = -2

Therefore, the point of intersection is (1, -2).

Conclusion

In this article, we used the graphical approach to solve a system of equations. We graphed each equation on a coordinate plane and found the point of intersection. The point of intersection is the solution to the system of equations.

Selecting the Correct Answer

Based on the point of intersection, we can select the correct answer from each drop-down menu.

Graph A: Correct
Graph B: Incorrect
Graph C: Incorrect
Graph D: Incorrect

Therefore, the correct answer is Graph A.

Discussion

The graphical approach is a useful method for solving systems of equations. It allows us to visualize the equations and find the point of intersection. However, it can be time-consuming and may not be as accurate as other methods.

References

[1] "Solving Systems of Equations" by Math Open Reference
[2] "Graphing Equations" by Khan Academy

Keywords

System of equations
Graphical approach
Point of intersection
Slope-intercept form
Coordinate plane
Algebra
Mathematics
Frequently Asked Questions (FAQs) about Solving Systems of Equations ====================================================================

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables.

Q: What is the graphical approach to solving systems of equations?

A: The graphical approach is a method of solving systems of equations by graphing each equation on a coordinate plane and finding the point of intersection.

Q: How do I graph an equation on a coordinate plane?

A: To graph an equation on a coordinate plane, you need to rewrite the equation in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. Then, you can plot the y-intercept and use the slope to find other points on the line.

Q: What is the point of intersection?

A: The point of intersection is the point where the two lines intersect. It is 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, you can substitute the x-coordinate into one of the equations and solve for y.

Q: What is the slope-intercept form of an equation?

A: The slope-intercept form of an equation is y = mx + b, where m is the slope and b is the y-intercept.

Q: What is the y-intercept?

A: The y-intercept is the point where the line intersects the y-axis. It is the value of y when x is equal to 0.

Q: How do I find the y-intercept?

A: To find the y-intercept, you need to rewrite the equation in slope-intercept form, y = mx + b, and then substitute x = 0 into the equation.

Q: What is the slope?

A: The slope is the ratio of the change in y to the change in x. It is a measure of how steep the line is.

Q: How do I find the slope?

A: To find the slope, you need to rewrite the equation in slope-intercept form, y = mx + b, and then look at the coefficient of x, which is the slope.

Q: What is the coordinate plane?

A: The coordinate plane is a grid of points that are used to graph equations. It is made up of two axes, the x-axis and the y-axis.

Q: How do I graph an equation on the coordinate plane?

A: To graph an equation on the coordinate plane, you need to rewrite the equation in slope-intercept form, y = mx + b, and then plot the y-intercept and use the slope to find other points on the line.

Q: What is algebra?

A: Algebra is a branch of mathematics that deals with the study of variables and their relationships. It is used to solve equations and inequalities.

Q: What is mathematics?

A: Mathematics is the study of numbers, quantities, and shapes. It is used to describe the world around us and to solve problems.

References

[1] "Solving Systems of Equations" by Math Open Reference
[2] "Graphing Equations" by Khan Academy
[3] "Algebra" by Wikipedia
[4] "Mathematics" by Wikipedia

Keywords

System of equations
Graphical approach
Point of intersection
Slope-intercept form
Coordinate plane
Algebra
Mathematics
Variables
Relationships
Equations
Inequalities