Find the solution to the system of equations: [tex]x+3 y=7[/tex] and [tex]2 x+4 y=8[/tex]
1. Isolate [tex]x[/tex] in the first equation:
2. Substitute the value for [tex]x[/tex] into the second equation:
3. Solve for [tex]y[/tex]:
[tex]
\begin{array}{l}
x=7-3 y \\
2(7-3 y)+4 y=8 \\
14-6 y+4 y=8 \\
14-2 y=8 \\
-2 y=-6 \\
y=3 \\
x+3(3)=7
\end{array}
[/tex]
4. Substitute [tex]y[/tex] into either original equation:
5. Write the solution as an ordered pair:
Answer (1)
Isolate x in the first equation: x = 7 − 3 y .
Substitute the expression for x into the second equation and solve for y : 2 ( 7 − 3 y ) + 4 y = 8 ⇒ y = 3 .
Substitute the value of y back into the equation for x : x = 7 − 3 ( 3 ) = − 2 .
Write the solution as an ordered pair: ( − 2 , 3 ) .
Explanation
Problem Analysis We are given a system of two linear equations: x + 3 y = 7 and 2 x + 4 y = 8 . Our goal is to find the values of x and y that satisfy both equations simultaneously. We will use the substitution method to solve this system.
Isolating x First, we solve the first equation for x in terms of y . This gives us: x = 7 − 3 y
Substitution Next, we substitute this expression for x into the second equation: 2 ( 7 − 3 y ) + 4 y = 8
Solving for y Now, we simplify and solve for y :
14 − 6 y + 4 y 14 − 2 y − 2 y − 2 y y y = 8 = 8 = 8 − 14 = − 6 = − 2 − 6 = 3
Solving for x Now that we have the value of y , we substitute it back into the equation x = 7 − 3 y to find the value of x :
x = 7 − 3 ( 3 ) = 7 − 9 = − 2
Final Answer Therefore, the solution to the system of equations is x = − 2 and y = 3 . We write the solution as an ordered pair ( − 2 , 3 ) .
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 traffic flow in a city. 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 12 cups of sugar available, we can set up a system of equations to determine how many cakes and pies the bakery can make. Let c be the number of cakes and p be the number of pies. The system of equations would be 2 c + p = 12 and c + 2 p = 12 . Solving this system would give the number of cakes and pies that can be made with the available ingredients.
