Solve the simultaneous equations:
$\begin{array}{l}
3 x+2 y=23 \\
4 x-y=16
\end{array}$
Answer (1)
Multiply the second equation by 2: 8 x − 2 y = 32 .
Add the modified second equation to the first equation: 11 x = 55 .
Solve for x : x = 5 .
Substitute the value of x back into the second equation and solve for y : y = 4 . The solution is x = 5 , y = 4 .
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: 3 x + 2 y = 23 Equation 2: 4 x − y = 16
Prepare for elimination We can use the method of elimination to solve this system. We'll multiply the second equation by 2 to make the coefficients of y in both equations opposites of each other.
Multiplying Equation 2 by 2, we get: 2 ( 4 x − y ) = 2 ( 16 ) 8 x − 2 y = 32
Now we have: Equation 1: 3 x + 2 y = 23 New Equation 2: 8 x − 2 y = 32
Eliminate y Now, we add the two equations to eliminate y :
( 3 x + 2 y ) + ( 8 x − 2 y ) = 23 + 32 11 x = 55
Solve for x Now, we solve for x :
11 x = 55 x = 11 55 x = 5
Substitute x into Equation 2 Now that we have the value of x , we can substitute it back into either of the original equations to solve for y . Let's use Equation 2: 4 x − y = 16 4 ( 5 ) − y = 16 20 − y = 16
Solve for y Now, we solve for y :
20 − y = 16 y = 20 − 16 y = 4
State the solution Therefore, the solution to the system of equations is x = 5 and y = 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. For example, suppose a bakery sells cakes and pies. Each cake requires 2 cups of flour and 1 cup of sugar, while each pie requires 1 cup of flour and 2 cups of sugar. If the bakery has 12 cups of flour and 9 cups of sugar available, we can set up a system of equations to determine how many cakes and pies the bakery can make. Let x be the number of cakes and y be the number of pies. The equations are 2 x + y = 12 and x + 2 y = 9 . Solving this system gives x = 5 and y = 2 , meaning the bakery can make 5 cakes and 2 pies.
