Consider the system of linear equations:
[tex]\begin{array}{l}
2 y=x+10 \\
3 y=3 x+15
\end{array}[/tex]
Which statements about the system are true? Check all that apply.
A. The system has one solution.
B. The system graphs parallel lines.
C. Both lines have the same slope.
D. Both lines have the same [tex]$y$[/tex]-intercept.
E. The equations graph the same line.
F. The solution is the intersection of the 2 lines.
Answer (2)
Rewrite the equations in slope-intercept form: y = 2 1 x + 5 and y = x + 5 .
Compare the slopes: 2 1 and 1 are different, so the lines are not parallel and the system has one solution.
Compare the y -intercepts: Both are 5 , so the lines have the same y -intercept.
Conclude that the system has one solution, the lines have the same y -intercept, and the solution is the intersection of the two lines. The system has one solution, Both lines have the same y -intercept, The solution is the intersection of the 2 lines.
Explanation
Understanding the Problem We are given a system of two linear equations:
2 y = x + 10 3 y = 3 x + 15
We need to determine which of the given statements about the system are true.
Strategy Let's rewrite both equations in slope-intercept form ( y = m x + b ), where m is the slope and b is the y -intercept. This will allow us to easily compare the properties of the two lines.
Analyzing the First Equation For the first equation, 2 y = x + 10 , we divide both sides by 2 to get:
y = 2 1 x + 5
So, the slope of the first line is 2 1 and the y -intercept is 5.
Analyzing the Second Equation For the second equation, 3 y = 3 x + 15 , we divide both sides by 3 to get:
y = x + 5
So, the slope of the second line is 1 and the y -intercept is 5.
Comparing Slopes and Intercepts Now, let's compare the slopes and y -intercepts of the two equations:
The slope of the first line is 2 1 , and the slope of the second line is 1. Since the slopes are different, the lines are not parallel and are not the same line.
The y -intercept of both lines is 5.
Determining the Correct Statements Since the slopes are different, the system has one solution, and the solution is the intersection of the two lines. The lines are not parallel, and they are not the same line. Both lines have the same y -intercept.
Conclusion Based on our analysis, the following statements are true:
The system has one solution.
Both lines have the same y -intercept.
The solution is the intersection of the 2 lines.
Examples
When you're trying to figure out where two roads intersect on a map, you're essentially solving a system of linear equations. Each road can be represented by a line, and the intersection point is the solution to the system. Similarly, in business, if you have two different cost equations for producing a product, finding the point where the costs are equal helps you determine the break-even point. This is a practical application of solving systems of equations to make informed decisions.
The system of equations has one solution, which is the intersection of the two lines, and both lines have the same y -intercept of 5. The lines are not parallel and do not have the same slope. Therefore, the true statements are A, D, and F.
;
