Solve the system of equations:
[tex]
\begin{array}{l}
y=-3 x+4 \\
-4 x+y=-10
\end{array}
[/tex]
A. (-6,22)
B. (1,1)
C. (2,-2)
D. No solution
Answer (2)
Substitute y = − 3 x + 4 into the second equation.
Solve for x : − 4 x + ( − 3 x + 4 ) = − 10 A rr x = 2 .
Substitute x = 2 back into the first equation: y = − 3 ( 2 ) + 4 A rry = − 2 .
The solution to the system of equations is ( 2 , − 2 ) .
Explanation
Analyze the problem We are given a system of two linear equations:
Equation 1: y = − 3 x + 4 Equation 2: − 4 x + y = − 10
Our goal is to find the values of x and y that satisfy both equations. We can use the substitution method to solve this system.
Substitution Substitute the expression for y from Equation 1 into Equation 2:
− 4 x + ( − 3 x + 4 ) = − 10
Solve for x Simplify and solve for x :
− 4 x − 3 x + 4 = − 10
− 7 x + 4 = − 10
− 7 x = − 14
x = 2
Solve for y Substitute the value of x back into Equation 1 to find y :
y = − 3 ( 2 ) + 4
y = − 6 + 4
y = − 2
Verify the solution So the solution is ( x , y ) = ( 2 , − 2 ) .
Let's check if this solution satisfies both equations:
Equation 1: − 2 = − 3 ( 2 ) + 4 ⇒ − 2 = − 6 + 4 ⇒ − 2 = − 2 (True)
Equation 2: − 4 ( 2 ) + ( − 2 ) = − 10 ⇒ − 8 − 2 = − 10 ⇒ − 10 = − 10 (True)
Since the solution satisfies both equations, it is the correct solution.
Final Answer The solution to the system of equations is ( 2 , − 2 ) .
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, a company might use a system of equations to determine the number of units they need to sell to cover their costs and start making a profit. By understanding how to solve systems of equations, you can make informed decisions in many practical situations.
The solution to the system of equations is (2, -2). This was found by substituting y from the first equation into the second, solving for x, and then back-substituting to find y. The solution checks out in both equations, confirming its validity.
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