Graph the following system of equations.
[tex]\begin{array}{l}
y=3 x+15 \\
3 x+3 y=9
\end{array}[/tex]
What is the solution to the system?
A. There is no solution.
B. There is one unique solution (-3,6).
C. There is one unique solution (0,15)
D. There are infinitely many solutions.
Answer (1)
Simplify the second equation to y = − x + 3 .
Recognize that the slopes of the two equations ( y = 3 x + 15 and y = − x + 3 ) are different, indicating a unique solution.
Set the two equations equal to each other and solve for x : 3 x + 15 = − x + 3 , which gives x = − 3 .
Substitute x = − 3 into one of the equations to solve for y , resulting in y = 6 . The solution is ( − 3 , 6 ) .
Explanation
Analyze the problem We are given a system of two linear equations:
y = 3 x + 15 3 x + 3 y = 9
Our goal is to find the solution to this system, which means finding the values of x and y that satisfy both equations. We also need to determine if there is no solution, a unique solution, or infinitely many solutions.
Simplify the equations First, let's simplify the second equation by dividing both sides by 3:
x + y = 3
Now, solve for y in the simplified second equation:
y = − x + 3
We now have two equations in slope-intercept form:
y = 3 x + 15 y = − x + 3
Comparing the two equations, we see that the slopes are different (3 and -1) and the y-intercepts are also different (15 and 3). Since the slopes are different, the lines intersect at one point, meaning there is a unique solution.
Find the solution To find the solution, we set the two equations equal to each other:
3 x + 15 = − x + 3
Now, solve for x :
3 x + x = 3 − 15 4 x = − 12 x = − 3
Substitute x = − 3 into either equation to find y . Let's use the second equation:
y = − ( − 3 ) + 3 y = 3 + 3 y = 6
So, the solution is x = − 3 and y = 6 , which is the point ( − 3 , 6 ) .
State the solution The solution to the system of equations is ( − 3 , 6 ) . This means that the lines intersect at the point ( − 3 , 6 ) .
Examples
Systems of equations are used in various real-world applications, such as determining the break-even point for a business. For example, if a company's cost function is y = 5 x + 1000 (where x is the number of units produced and y is the total cost) and its revenue function is y = 15 x , solving this system of equations will give the number of units the company needs to sell to break even. In this case, solving 5 x + 1000 = 15 x gives x = 100 , meaning the company needs to sell 100 units to break even.
