Solve the system with the addition method:
$\left\{\begin{array}{ll}
4 x+10 y & =68 \\
x+8 y & =39
\end{array}\right.$
Answer: $(x, y)=($ $\square$ , $\square$ )
Enter your answers as integers or as reduced fraction(s) in the form A/B.
Answer (2)
The solution to the system of equations is (7, 4). We eliminated one variable and solved for the other by substituting back into one of the original equations. This method effectively helps us find the values of both variables in the system.
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Multiply the second equation by -4 to eliminate x.
Add the equations to eliminate x and solve for y: y = 4 .
Substitute the value of y into one of the original equations to solve for x: x = 7 .
The solution to the system of equations is ( 7 , 4 ) .
Explanation
Analyze the problem We are given a system of two linear equations:
{ 4 x + 10 y = 68 x + 8 y = 39
Our goal is to solve this system for x and y using the addition method.
Eliminate x To eliminate x , we can multiply the second equation by − 4 :
− 4 ( x + 8 y ) = − 4 ( 39 )
− 4 x − 32 y = − 156
Now our system of equations is:
{ 4 x + 10 y = 68 − 4 x − 32 y = − 156
Solve for y Add the two equations to eliminate x :
( 4 x + 10 y ) + ( − 4 x − 32 y ) = 68 + ( − 156 )
− 22 y = − 88
Divide both sides by − 22 to solve for y :
y = − 22 − 88 = 4
So, y = 4 .
Solve for x Substitute the value of y back into one of the original equations to solve for x . Let's use the second equation:
x + 8 y = 39
x + 8 ( 4 ) = 39
x + 32 = 39
x = 39 − 32 = 7
So, x = 7 .
Final Answer The solution to the system of equations is ( x , y ) = ( 7 , 4 ) .
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 supply and demand in economics. In this case, imagine you're buying items at a store. The equations could represent the total cost of different combinations of items, and solving the system helps you find the individual prices of each item. For example, if 4 apples and 10 bananas cost $68, and 1 apple and 8 bananas cost $39, solving the system tells you the price of one apple and one banana.
