Solve the simultaneous equations
$\begin{aligned}
3 x & =9+y \\
x+5 y & =5
\end{aligned}$
Answer (1)
Solve the first equation for y : y = 3 x − 9 .
Substitute this expression for y into the second equation: x + 5 ( 3 x − 9 ) = 5 .
Solve for x : x = 8 25 .
Substitute the value of x back into the equation for y : y = 8 3 .
The solution is x = 8 25 , y = 8 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: 3 x = 9 + y Equation 2: x + 5 y = 5
Solve for y in Equation 1 We can solve this system of equations using the substitution or elimination method. Let's use the substitution method. First, we solve Equation 1 for y :
3 x = 9 + y y = 3 x − 9
Substitute into Equation 2 Now, substitute this expression for y into Equation 2:
x + 5 y = 5 x + 5 ( 3 x − 9 ) = 5
Solve for x Simplify and solve for x :
x + 15 x − 45 = 5 16 x = 50 x = 16 50 = 8 25
Solve for y Now that we have the value of x , we can substitute it back into the expression for y :
y = 3 x − 9 y = 3 ( 8 25 ) − 9 y = 8 75 − 8 72 = 8 3
State the solution Therefore, the solution to the system of equations is x = 8 25 and y = 8 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 cookies and cakes. Each cookie requires 0.1 kg of flour and 0.05 kg of sugar, while each cake requires 0.5 kg of flour and 0.2 kg of sugar. If the bakery has 5 kg of flour and 2 kg of sugar available, we can set up a system of equations to determine how many cookies and cakes the bakery can make. Solving this system helps the bakery optimize its production based on available resources.
