\begin{aligned} 3 y+11 & =4 x \\ 10 x+2 y+1 & =0\end{aligned}
Answer (1)
Rewrite the equations in standard form: 4 x − 3 y = 11 and 10 x + 2 y = − 1 .
Use the elimination method to eliminate y by multiplying the first equation by 2 and the second equation by 3.
Add the modified equations to solve for x : 38 x = 19 , so x = 2 1 .
Substitute x = 2 1 into one of the original equations to solve for y : y = − 3 .
The solution to the system of equations is x = 2 1 , y = − 3 .
Explanation
Problem Setup We are given a system of two linear equations:
Equation 1: 3 y + 11 = 4 x
Equation 2: 10 x + 2 y + 1 = 0
Our goal is to find the values of x and y that satisfy both equations.
Rewriting Equations Let's rewrite the equations in the standard form A x + B y = C :
Equation 1: 4 x − 3 y = 11 Equation 2: 10 x + 2 y = − 1
Elimination Method We can use the elimination method to solve this system. Multiply the first equation by 2 and the second equation by 3 to eliminate y :
Equation 1 (multiplied by 2): 8 x − 6 y = 22 Equation 2 (multiplied by 3): 30 x + 6 y = − 3
Adding Equations Add the two modified equations to eliminate y :
( 8 x − 6 y ) + ( 30 x + 6 y ) = 22 + ( − 3 ) 38 x = 19
Solving for x Solve for x :
x = 38 19 = 2 1
Solving for y Substitute the value of x back into one of the original equations to solve for y . Let's use Equation 1:
4 ( 2 1 ) − 3 y = 11 2 − 3 y = 11 − 3 y = 9 y = − 3
Final Solution Therefore, the solution to the system of equations is x = 2 1 and y = − 3 .
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 ingredients in a recipe, or modeling traffic flow in a city. Understanding how to solve systems of equations allows us to find solutions to problems involving multiple variables and constraints. For instance, a company might use a system of equations to determine the number of units they need to sell to cover their costs and start making a profit. By setting up equations that represent their revenue and expenses, they can find the point where these two lines intersect, indicating the break-even point.
