Solve the following system of equations.
$\begin{array}{l}
8 x+3 y=-19 \\
7 x-4 y=-10
\end{array}$
Answer (1)
Multiply equations to make the coefficients of y opposites.
Add the equations to eliminate y and solve for x .
Substitute the value of x back into one of the original equations and solve for y .
The solution to the system of equations is x = − 2 , y = − 1 .
Explanation
Analyzing the problem We are given a system of two linear equations with two variables x and y:
Equation 1: 8 x + 3 y = − 19 Equation 2: 7 x − 4 y = − 10
The goal is to find the values of x and y that satisfy both equations.
Eliminating y We can solve this system of equations using the elimination method. Multiply the first equation by 4 and the second equation by 3 to eliminate y.
This gives us: 4 ( 8 x + 3 y ) = 4 ( − 19 ) ⇒ 32 x + 12 y = − 76 3 ( 7 x − 4 y ) = 3 ( − 10 ) ⇒ 21 x − 12 y = − 30
Adding the equations Add the two equations to eliminate y: ( 32 x + 12 y ) + ( 21 x − 12 y ) = − 76 + ( − 30 ) 32 x + 21 x + 12 y − 12 y = − 106 53 x = − 106
Solving for x Solve for x: x = 53 − 106 = − 2
Substituting x into Equation 1 Substitute the value of x back into either of the original equations to solve for y. Using the first equation: 8 ( − 2 ) + 3 y = − 19 − 16 + 3 y = − 19
Solving for y Solve for y: 3 y = − 19 + 16 3 y = − 3 y = 3 − 3 = − 1
Final Answer Therefore, the solution to the system of equations is x = − 2 and y = − 1 .
Examples
Systems of equations are used in various fields such as economics, engineering, and computer science. For example, in economics, they can be used to model the supply and demand curves in a market. The intersection point of these curves represents the equilibrium price and quantity. Similarly, in electrical engineering, systems of equations can be used to analyze circuits and determine the current and voltage in different parts of the circuit. Understanding how to solve systems of equations is fundamental for solving real-world problems in these fields.
