Solve the system of linear equations:
[tex]
\begin{array}{l}
x-y=2 \
3 x+5 y=46
\end{array}
[/tex]
x=
y=
Answer (2)
Solve the first equation for x : x = y + 2 .
Substitute this expression for x into the second equation: 3 ( y + 2 ) + 5 y = 46 .
Simplify and solve for y : y = 5 .
Substitute the value of y back into the equation x = y + 2 to find x : x = 7 .
The solution is x = 7 , y = 5 .
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 given equations are:
Equation 1: x − y = 2 Equation 2: 3 x + 5 y = 46
Solve for x in the first equation We can solve this system of equations using the substitution method. First, solve Equation 1 for x :
x = y + 2
Substitute x into the second equation Now, substitute this expression for x into Equation 2:
3 ( y + 2 ) + 5 y = 46
Solve for y Simplify and solve for y :
3 y + 6 + 5 y = 46
Combine like terms:
8 y + 6 = 46
Subtract 6 from both sides:
8 y = 40
Divide by 8:
y = 5
Solve for x Now that we have the value of y , we can substitute it back into the equation x = y + 2 to find x :
x = 5 + 2
x = 7
State the solution Therefore, the solution to the system of equations is x = 7 and y = 5 .
Examples
Systems of linear 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. Understanding how to solve these systems allows us to make informed decisions and solve practical problems efficiently. For example, if a company sells two products, we can use a system of equations to determine how many units of each product they need to sell to reach a certain profit target. The solution x = 7 and y = 5 means that the company needs to sell 7 units of the first product and 5 units of the second product to reach the target.
The solution to the system of equations is x = 7 and y = 5 .
;
