Solve the system for $x$ and $y$.
$\begin{array}{l}
-8 x-y=-43 \\
0-5 y=-71
\end{array}$
A. $(3,19)$
B. $(-2,11)$
C. $(2.27)$
D. (4.11)
Answer (2)
The solution to the system of equations is x = 3.6 and y = 14.2 . None of the provided multiple-choice answers match this solution exactly, indicating a possible mistake in the choices. The correct solution indicates values of approximately ( 3.6 , 14.2 ) .
;
Solve the second equation for y : y = 5 71 = 14.2 .
Substitute the value of y into the first equation: − 8 x − 5 71 = − 43 .
Solve for x : x = 5 18 = 3.6 .
The solution to the system of equations is x = 3.6 and y = 14.2 .
Explanation
Analyze the problem We are given a system of two linear equations with two variables, x and y . Our goal is to find the values of x and y that satisfy both equations.
The given equations are:
− 8 x − y = − 43 0 − 5 y = − 71
We will solve this system using substitution.
Solve for y First, let's solve the second equation for y :
− 5 y = − 71 Divide both sides by − 5 :
y = − 5 − 71 = 5 71 So, y = 5 71 = 14.2 .
Solve for x Now, substitute the value of y into the first equation: − 8 x − y = − 43 − 8 x − 5 71 = − 43 Add 5 71 to both sides: − 8 x = − 43 + 5 71 To add these numbers, we need a common denominator, which is 5: − 8 x = 5 − 43 × 5 + 5 71 − 8 x = 5 − 215 + 5 71 − 8 x = 5 − 215 + 71 − 8 x = 5 − 144 Now, divide both sides by − 8 :
x = 5 − 144 ÷ − 8 x = 5 − 144 × − 8 1 x = − 40 − 144 x = 40 144 Simplify the fraction by dividing both numerator and denominator by 8: x = 5 18 So, x = 5 18 = 3.6 .
Find the solution Therefore, the solution to the system of equations is x = 3.6 and y = 14.2 .
Now, let's compare our solution ( 3.6 , 14.2 ) with the given options: a. ( 3 , 19 ) b. ( − 2 , 11 ) c. ( 2.27 ) d. ( 4 , 11 )
None of the options match our solution exactly. However, option 'c' is closest to the x value. It seems there might be a typo in the options. The correct answer should be close to (3.6, 14.2).
Final Answer The solution to the system of equations is x = 5 18 = 3.6 and y = 5 71 = 14.2 .
Examples
Systems of equations are used in various real-life situations, such as determining the break-even point for a business. For example, a company might use a system of equations to model its costs and revenues, and then solve the system to find the production level at which costs equal revenues. This helps the company make informed decisions about pricing and production levels. Another example is in electrical engineering, where systems of equations are used to analyze circuits and determine the currents and voltages at different points in the circuit.
