Solve $3 x-2 y=10$ and $x+3 y=7$.
Answer (1)
Solve the second equation for x : x = 7 − 3 y .
Substitute this expression for x into the first equation and solve for y : 3 ( 7 − 3 y ) − 2 y = 10 A rry = 1 .
Substitute the value of y back into the expression for x : x = 7 − 3 ( 1 ) A rr x = 4 .
The solution to the system of equations is x = 4 , y = 1 .
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:
3 x − 2 y = 10
x + 3 y = 7
We can solve this system using either the substitution method or the elimination method. Here, we will use the substitution method.
Solve for x in the second equation First, we solve the second equation for x :
x + 3 y = 7 x = 7 − 3 y
Substitute into the first equation Now, we substitute this expression for x into the first equation:
3 x − 2 y = 10 3 ( 7 − 3 y ) − 2 y = 10
Solve for y Next, we simplify and solve for y :
21 − 9 y − 2 y = 10 21 − 11 y = 10 − 11 y = 10 − 21 − 11 y = − 11 y = − 11 − 11 y = 1
Solve for x Now that we have the value of y , we can substitute it back into the expression for x :
x = 7 − 3 y x = 7 − 3 ( 1 ) x = 7 − 3 x = 4
State the solution Therefore, the solution to the system of equations is x = 4 and y = 1 .
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 c be the number of cakes and p be the number of pies. Then, the system of equations is 2 c + p = 12 and c + 2 p = 9 . Solving this system gives c = 5 and p = 2 , meaning the bakery can make 5 cakes and 2 pies.
