What is the solution to this system of linear equations?
$\begin{array}{l}
2 x+3 y=3 \\
7 x-3 y=24
\end{array}$
A. $(2,7)$
B. $(3,-21)$
C. $(3,-1)$
D. $(9,0)$
Answer (1)
Substitute each given option into the system of equations.
Check if both equations are satisfied for each option.
Option (3, -1) satisfies both equations: 2 ( 3 ) + 3 ( − 1 ) = 3 and 7 ( 3 ) − 3 ( − 1 ) = 24 .
The solution to the system of equations is ( 3 , − 1 ) .
Explanation
Analyze the problem We are given a system of two linear equations:
2 x + 3 y = 3 7 x − 3 y = 24
We need to find the solution (x, y) from the given options: (2, 7), (3, -21), (3, -1), (9, 0).
Test each option Let's test each option to see which one satisfies both equations.
Option 1: (2, 7) Equation 1: 2 ( 2 ) + 3 ( 7 ) = 4 + 21 = 25 . This is not equal to 3, so (2, 7) is not a solution. Equation 2: 7 ( 2 ) − 3 ( 7 ) = 14 − 21 = − 7 . This is not equal to 24, so (2, 7) is not a solution.
Option 2: (3, -21) Equation 1: 2 ( 3 ) + 3 ( − 21 ) = 6 − 63 = − 57 . This is not equal to 3, so (3, -21) is not a solution. Equation 2: 7 ( 3 ) − 3 ( − 21 ) = 21 + 63 = 84 . This is not equal to 24, so (3, -21) is not a solution.
Option 3: (3, -1) Equation 1: 2 ( 3 ) + 3 ( − 1 ) = 6 − 3 = 3 . This satisfies the first equation. Equation 2: 7 ( 3 ) − 3 ( − 1 ) = 21 + 3 = 24 . This satisfies the second equation.
Option 4: (9, 0) Equation 1: 2 ( 9 ) + 3 ( 0 ) = 18 + 0 = 18 . This is not equal to 3, so (9, 0) is not a solution. Equation 2: 7 ( 9 ) − 3 ( 0 ) = 63 − 0 = 63 . This is not equal to 24, so (9, 0) is not a solution.
Find the solution Only option (3, -1) satisfies both equations. Therefore, the solution to the system of linear equations is (3, -1).
Examples
Systems of linear equations are used in various real-life applications, such as determining the optimal mix of products in manufacturing, balancing chemical equations, and modeling supply and demand in economics. 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 their production costs. Solving these systems helps in making informed decisions and optimizing outcomes.
