Find the solution for this system of equations:
[tex]\begin{aligned}
12 x+15 y & =34 \
-6 x+5 y & =3
\end{aligned}[/tex]
x = [ ]
y = [ ]
Answer (2)
Multiply the second equation by 2 to make the coefficients of x opposites.
Add the modified second equation to the first equation to eliminate x and solve for y .
Substitute the value of y back into one of the original equations to solve for x .
The solution to the system of equations is x = 6 5 , y = 5 8 .
Explanation
Analyzing the System of Equations We are given a system of two linear equations in two variables x and y :
12 x + 15 y = 34 − 6 x + 5 y = 3
Our goal is to find the values of x and y that satisfy both equations simultaneously. We can use the method of elimination to solve this system.
Preparing for Elimination To eliminate x , we can multiply the second equation by 2:
2 × ( − 6 x + 5 y ) = 2 × 3
This gives us:
− 12 x + 10 y = 6
Now we have the following system:
12 x + 15 y = 34 − 12 x + 10 y = 6
Eliminating x and Solving for y Add the two equations to eliminate x :
( 12 x + 15 y ) + ( − 12 x + 10 y ) = 34 + 6
This simplifies to:
25 y = 40
Now, solve for y :
y = 25 40 = 5 8
So, y = 5 8 .
Solving for x Substitute the value of y back into one of the original equations to solve for x . Let's use the second equation:
− 6 x + 5 y = 3
Substitute y = 5 8 :
− 6 x + 5 ( 5 8 ) = 3
Simplify:
− 6 x + 8 = 3
Subtract 8 from both sides:
− 6 x = 3 − 8
− 6 x = − 5
Solve for x :
x = − 6 − 5 = 6 5
So, x = 6 5 .
Final Answer Therefore, the solution to the system of equations is:
x = 6 5 , y = 5 8
We can write the solution as x = 6 5 and y = 5 8 .
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 products to maximize profit, or modeling supply and demand in economics. For instance, if a company sells two products, the system of equations can help determine the number of units of each product that need to be sold to cover all costs and achieve a desired profit level. Understanding how to solve these systems is crucial for making informed decisions in business and economics.
The solution to the system of equations is x = 6 5 and y = 5 8 . We used the elimination method to solve for the variables. After transforming the second equation and adding, we found the value of y and subsequently found x.
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