What is the solution to the system of equations below?
[tex]y=-\frac{1}{4} x+2[/tex] and [tex]3 y=-\frac{3}{4} x-6[/tex]
A. no solution
B. infinitely many solutions
C. [tex](-16,6)[/tex]
D. [tex](-16,-2)[/tex]
Answer (2)
The system of equations has no solution because the two equations represent parallel lines that will never intersect. This is determined by analyzing the slopes and y-intercepts of the equations. Thus, the correct answer is option A: no solution.
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Multiply the first equation by 3: 3 y = − 4 3 x + 6 .
Compare the modified first equation with the second equation: 3 y = − 4 3 x − 6 .
Observe that the left-hand sides are the same, but the right-hand sides are different, indicating parallel lines.
Conclude that there is no solution to the system of equations: no solution .
Explanation
Analyze the problem We are given the system of equations:
y = − 4 1 x + 2 3 y = − 4 3 x − 6
We want to find the solution to this system.
Transform the first equation First, let's multiply the first equation by 3:
3 ( y ) = 3 ( − 4 1 x + 2 ) 3 y = − 4 3 x + 6
Now we have two equations:
3 y = − 4 3 x + 6 3 y = − 4 3 x − 6
Compare the equations Comparing the two equations, we see that the left-hand sides are the same ( 3 y ), but the right-hand sides are different ( − 4 3 x + 6 and − 4 3 x − 6 ). This means that the two lines have the same slope but different y-intercepts, so they are parallel and will never intersect. Therefore, there is no solution to the system of equations.
Conclusion The system of equations has no solution.
Examples
Systems of equations are used in many real-world applications, such as determining the break-even point for a business. For example, if you have a business and want to know how many products you need to sell to make your revenue equal to your costs, you can set up a system of equations. One equation could represent your revenue (price per product times the number of products sold), and the other equation could represent your costs (fixed costs plus variable costs per product times the number of products sold). Solving this system of equations will give you the number of products you need to sell to break even.
