Solve the system of equations:
[tex]$\begin{array}{l}
4 x+3 y=8 \\
4 x-3 y=-16
\end{array}$[/tex]
Answer (2)
The solution to the system of equations is x = − 1 and y = 4 . We used the elimination method to first eliminate y and solve for x before substituting back to find y . The final answer is x = − 1 , y = 4 .
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Use the elimination method to eliminate y by adding the two equations.
Solve for x : 8 x = − 8 , so x = − 1 .
Substitute x = − 1 into the first equation: 4 ( − 1 ) + 3 y = 8 .
Solve for y : 3 y = 12 , so y = 4 . The solution is x = − 1 , y = 4 .
Explanation
Understanding 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.
Setting up the Elimination Method We have the following system of equations:
Equation 1: 4 x + 3 y = 8
Equation 2: 4 x − 3 y = − 16
We can use the method of elimination to solve this system. Notice that the coefficients of y in the two equations are 3 and − 3 , which are opposites. This makes it easy to eliminate y by adding the two equations.
Eliminating y and Solving for x Add Equation 1 and Equation 2: ( 4 x + 3 y ) + ( 4 x − 3 y ) = 8 + ( − 16 )
Simplify the equation: 8 x = − 8
Finding the Value of x Divide both sides of the equation by 8 to solve for x :
8 8 x = 8 − 8 x = − 1
Substituting x into Equation 1 Now that we have the value of x , we can substitute it into either Equation 1 or Equation 2 to solve for y . Let's use Equation 1: 4 x + 3 y = 8
Substitute x = − 1 :
4 ( − 1 ) + 3 y = 8 − 4 + 3 y = 8
Isolating y Add 4 to both sides of the equation: − 4 + 3 y + 4 = 8 + 4 3 y = 12
Finding the Value of y Divide both sides of the equation by 3 to solve for y :
3 3 y = 3 12 y = 4
Stating the Solution Therefore, the solution to the system of equations is x = − 1 and y = 4 .
Examples
Systems of equations are used in various real-life scenarios, 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 100 cups of flour and 80 cups of sugar available, we can set up a system of equations to determine how many cakes and pies the bakery can make to use all the available ingredients. Solving this system helps the bakery optimize its production.
