What is the solution $(q, r)$ to this system of linear equations?
$\begin{array}{l}
12 q+3 r=15 \\
-4 q-4 r=-44
\end{array}$
A. $(-18,29)$
B. $(-2,13)$
C. $(8,-1)$
D. $(15,-44)$
Answer (2)
Simplify the given equations: 12 q + 3 r = 15 becomes 4 q + r = 5 , and − 4 q − 4 r = − 44 becomes q + r = 11 .
Solve for r in the second equation: r = 11 − q .
Substitute the expression for r into the first equation and solve for q : 4 q + ( 11 − q ) = 5 , which gives q = − 2 .
Substitute the value of q back into the equation for r : r = 11 − ( − 2 ) = 13 . The solution is ( − 2 , 13 ) .
Explanation
Analyze the problem We are given a system of two linear equations with two variables, q and r . Our goal is to find the values of q and r that satisfy both equations simultaneously. The given equations are:
12 q + 3 r = 15 − 4 q − 4 r = − 44
We will use the substitution or elimination method to solve this system.
Simplify the equations First, let's simplify both equations to make the calculations easier. We can divide the first equation by 3 and the second equation by -4:
3 12 q + 3 r = 3 15 ⇒ 4 q + r = 5 − 4 − 4 q − 4 r = − 4 − 44 ⇒ q + r = 11
So, our simplified equations are:
4 q + r = 5 q + r = 11
Solve for r in terms of q Now, we can solve for r in the second equation:
r = 11 − q
Substitute into the first equation Substitute this expression for r into the first simplified equation:
4 q + ( 11 − q ) = 5
Solve for q Now, solve for q :
4 q + 11 − q = 5 ⇒ 3 q = 5 − 11 ⇒ 3 q = − 6 ⇒ q = 3 − 6 ⇒ q = − 2
Solve for r Substitute the value of q back into the equation r = 11 − q to find r :
r = 11 − ( − 2 ) = 11 + 2 = 13
State the solution So, the solution is q = − 2 and r = 13 . Therefore, the solution to the system of equations is ( q , r ) = ( − 2 , 13 ) .
Verify the solution To verify the solution, substitute q = − 2 and r = 13 into the original equations:
12 q + 3 r = 12 ( − 2 ) + 3 ( 13 ) = − 24 + 39 = 15 − 4 q − 4 r = − 4 ( − 2 ) − 4 ( 13 ) = 8 − 52 = − 44
Both equations are satisfied, so our solution is correct.
Thus, the solution to the system of equations is ( − 2 , 13 ) .
Examples
Systems of linear equations are used in various real-world applications, such as determining the optimal mix of products to maximize profit, balancing chemical equations, and solving network flow problems. For example, a company might use a system of equations to determine how many units of two different products they need to sell to reach a specific revenue target, given the prices of the products and the costs involved. Understanding how to solve these systems is crucial for making informed decisions in many fields.
The solution to the system of linear equations is (q, r) = (-2, 13). This was found by simplifying the equations, isolating one variable, and substituting it back into the other equation. The chosen option is B: (-2, 13).
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