Select the correct answer.
Consider the following system of equations.
[tex]\begin{aligned}
-2 x+5 y & =19 \\
y & =-\frac{5}{6} x-\frac{1}{6}
\end{aligned}[/tex]
Use this graph of the system to approximate its solution.
A. [tex]\left(\frac{5}{2},-\frac{13}{4}\right)[/tex]
B. [tex]\left(-\frac{13}{4}, \frac{5}{2}\right)[/tex]
C. [tex]\left(\frac{13}{4},-\frac{5}{2}\right)[/tex]
D. [tex]\left(-\frac{5}{2}, \frac{13}{4}\right)[/tex]
Answer (2)
Substitute each option into the system of equations.
Check which option satisfies both equations.
Option A: ( 2 5 , − 4 13 ) does not satisfy the first equation.
Option B: ( − 4 13 , 2 5 ) satisfies both equations.
Option C: ( 4 13 , − 2 5 ) does not satisfy the first equation.
Option D: ( − 2 5 , 4 13 ) does not satisfy the first equation.
The approximate solution is ( − 4 13 , 2 5 ) .
Explanation
Analyze the problem and equations We are given a system of two equations:
− 2 x + 5 y = 19 y = − 6 5 x − 6 1
We need to find the correct option that satisfies both equations. We will substitute each option into the equations to see which one works.
Check option A Let's check option A: ( 2 5 , − 4 13 )
Equation 1: − 2 ( 2 5 ) + 5 ( − 4 13 ) = − 5 − 4 65 = 4 − 20 − 65 = 4 − 85 = − 21.25 = 19
Since the first equation is not satisfied, we don't need to check the second equation.
Check option B Let's check option B: ( − 4 13 , 2 5 )
Equation 1: − 2 ( − 4 13 ) + 5 ( 2 5 ) = 2 13 + 2 25 = 2 38 = 19 Equation 2: 2 5 = − 6 5 ( − 4 13 ) − 6 1 = 24 65 − 6 1 = 24 65 − 4 = 24 61 ≈ 2.54 = 2.5
Since the first equation is satisfied, but the second equation is approximately satisfied (given that we are approximating the solution from a graph), this could be the correct answer.
Check option C Let's check option C: ( 4 13 , − 2 5 )
Equation 1: − 2 ( 4 13 ) + 5 ( − 2 5 ) = − 2 13 − 2 25 = − 2 38 = − 19 = 19
Since the first equation is not satisfied, we don't need to check the second equation.
Check option D Let's check option D: ( − 2 5 , 4 13 )
Equation 1: − 2 ( − 2 5 ) + 5 ( 4 13 ) = 5 + 4 65 = 4 20 + 65 = 4 85 = 21.25 = 19
Since the first equation is not satisfied, we don't need to check the second equation.
Conclusion From the calculations, option B ( − 4 13 , 2 5 ) is the closest to satisfying both equations. Therefore, the approximate solution is ( − 4 13 , 2 5 ) .
Examples
Systems of equations are used in various real-world applications, such as determining the break-even point for a business, calculating the optimal mix of products to maximize profit, or modeling supply and demand in economics. In engineering, systems of equations are used to analyze circuits, design structures, and model fluid flow. Understanding how to solve systems of equations is crucial for making informed decisions and solving complex problems in many fields.
The approximate solution to the system of equations based on the checks performed is option B: ( − 4 13 , 2 5 ) . This option satisfies the first equation perfectly and is a close approximation for the second equation. The other options do not meet the criteria set by the equations.
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