Solve the system of equations:

[tex]\left\{\begin{array}{l}
y=-4 x+4 \
-4 x+5 y=-28
\end{array}\right.[/tex]

Solution: ( [tex]\square[/tex] , [tex]\square[/tex] )

The solution to the system of equations is (2, -4), found using substitution and verifying that it satisfies both original equations. The process involved isolating y in the first equation and substituting into the second equation. After solving, we confirmed the solution by substituting back to check both equations.
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Answered by Anonymous | 2025-07-04

Substitute the expression for y from the first equation into the second equation.
Solve the resulting equation for x : x = 2 .
Substitute the value of x back into the first equation to find y : y = − 4 .
The solution to the system of equations is ( 2 , − 4 ) ​ .

Explanation

Analyze the problem We are given a system of two linear equations:

Equation 1: y = − 4 x + 4 Equation 2: − 4 x + 5 y = − 28
Our goal is to find the values of x and y that satisfy both equations.

Substitution We will use the substitution method. Since Equation 1 already expresses y in terms of x , we can substitute this expression into Equation 2:

− 4 x + 5 ( − 4 x + 4 ) = − 28

Solve for x Now, we simplify and solve for x :

− 4 x − 20 x + 20 = − 28 − 24 x = − 48 x = − 24 − 48 ​ x = 2

Solve for y Now that we have the value of x , we can substitute it back into Equation 1 to find the value of y :

y = − 4 ( 2 ) + 4 y = − 8 + 4 y = − 4

Verification So, the solution to the system of equations is x = 2 and y = − 4 . We can write this as an ordered pair ( 2 , − 4 ) .

Let's verify the solution by substituting these values into both equations:
Equation 1: − 4 = − 4 ( 2 ) + 4 ⇒ − 4 = − 8 + 4 ⇒ − 4 = − 4 (True) Equation 2: − 4 ( 2 ) + 5 ( − 4 ) = − 28 ⇒ − 8 − 20 = − 28 ⇒ − 28 = − 28 (True)
Since the solution satisfies both equations, it is correct.

Final Answer The solution to the system of equations is ( 2 , − 4 ) .

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 company produces two products, A and B. Each product requires a certain amount of labor and materials. By setting up a system of equations, the company can determine the optimal production levels for each product to maximize profit, given the constraints on labor and materials. Let's say product A requires 2 hours of labor and 3 units of material, while product B requires 3 hours of labor and 1 unit of material. If the company has 20 hours of labor and 15 units of material available, the system of equations would be:
2 x + 3 y = 20 (labor constraint) 3 x + y = 15 (material constraint)
where x is the number of units of product A and y is the number of units of product B. Solving this system would give the optimal production levels for each product.

Answered by GinnyAnswer | 2025-07-04