$x+5 y=46$ and $x-y=-14$
Answer (2)
Solve the second equation for x : x = y − 14 .
Substitute this expression into the first equation and solve for y : y = 10 .
Substitute the value of y back into the expression for x : x = − 4 .
The solution to the system of equations is x = − 4 , y = 10 .
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: x + 5 y = 46 Equation 2: x − y = − 14
Solve for x in terms of y We can use the substitution method to solve this system. First, we solve the second equation for x in terms of y :
x − y = − 14 x = y − 14
Substitute into the first equation Now, we substitute this expression for x into the first equation:
( y − 14 ) + 5 y = 46
Solve for y Next, we simplify and solve for y :
y − 14 + 5 y = 46 6 y − 14 = 46 6 y = 46 + 14 6 y = 60 y = 6 60 y = 10
Solve for x Now that we have the value of y , we can substitute it back into the expression for x :
x = y − 14 x = 10 − 14 x = − 4
State the solution Therefore, the solution to the system of equations is x = − 4 and y = 10 .
Examples
Systems of equations are used in various real-world applications. For example, they can be used to model supply and demand in economics, determine the optimal mix of ingredients in manufacturing, or analyze electrical circuits. In this case, imagine you're buying fruits at a store. Apples cost x each, and bananas cost y each. If you buy one apple and five bananas for $46, and one apple minus one banana costs you -$14 (meaning you get $14 back), you can use this system of equations to find the individual prices of apples and bananas.
The solution to the system of equations x + 5 y = 46 and x − y = − 14 is x = − 4 and y = 10 . We solved for x in terms of y and substituted this value into the first equation. This method is a clear way to find values that satisfy both equations.
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