Solve this system of linear equations. Write your answer as an ordered pair, $(x, y)$.
$\begin{array}{l}
6 x+4 y=24 \\
6 x+3 y=21
\end{array}$
Answer (1)
Subtract the second equation from the first to eliminate x : y = 3 .
Substitute y = 3 into the second equation: 6 x + 3 ( 3 ) = 21 .
Solve for x : x = 2 .
Express the solution as an ordered pair: ( 2 , 3 ) .
Explanation
Analyze the problem We are given a system of two linear equations with two variables, x and y . Our goal is to find the values of x and y that satisfy both equations simultaneously. The equations are:
Equation 1: 6 x + 4 y = 24 Equation 2: 6 x + 3 y = 21
Eliminate x To solve this system, we can use the method of elimination. We notice that the coefficient of x is the same in both equations, which makes it easy to eliminate x . We subtract Equation 2 from Equation 1:
( 6 x + 4 y ) − ( 6 x + 3 y ) = 24 − 21
Solve for y Simplifying the equation, we get:
6 x + 4 y − 6 x − 3 y = 3
y = 3
Substitute y into Equation 2 Now that we have the value of y , we can substitute it back into either Equation 1 or Equation 2 to solve for x . Let's substitute y = 3 into Equation 2:
6 x + 3 ( 3 ) = 21
6 x + 9 = 21
Isolate x Subtract 9 from both sides:
6 x = 21 − 9
6 x = 12
Solve for x Divide by 6:
x = 6 12
x = 2
State the solution Therefore, the solution to the system of equations is x = 2 and y = 3 . We write the solution as an ordered pair ( x , y ) = ( 2 , 3 ) .
Examples
Imagine you're buying apples and bananas. The first time, you buy 6 apples and 4 bananas for $24. The second time, you buy 6 apples and 3 bananas for $21. By solving this system of equations, you can find the price of one apple ($2) and one banana ($3). This method is useful in many real-life situations where you have multiple unknowns and multiple pieces of information relating them.
