Solve the system of equations:
[tex]
\begin{array}{l}
y=-2y+6 \
-4y+y=-20
\end{array}
[/tex]
Answer (2)
Solve the first equation for y : y = − 2 y + 6 ⇒ y = 2 .
Substitute the value of y into the second equation: − 4 x + y = − 20 ⇒ − 4 x + 2 = − 20 .
Solve for x : − 4 x = − 22 ⇒ x = 2 11 = 5.5 .
The solution to the system of equations is ( 2 11 , 2 ) .
Explanation
Understanding the Problem We are given a system of two equations with two variables, x and y . The equations are:
y = − 2 y + 6
− 4 x + y = − 20
The objective is to solve for x and y .
The possible solutions are given as (4, -4), (6, -4), (2, -12), and none of these. Note that the options provided in the original problem statement are incorrect, I will solve the problem and provide the correct solution.
Solving for y First, let's solve the first equation for y :
y = − 2 y + 6
Add 2 y to both sides:
y + 2 y = 6
3 y = 6
Divide both sides by 3:
y = 3 6
y = 2
Solving for x Now, substitute the value of y into the second equation:
− 4 x + y = − 20
− 4 x + 2 = − 20
Subtract 2 from both sides:
− 4 x = − 20 − 2
− 4 x = − 22
Divide both sides by -4:
x = − 4 − 22
x = 2 11
So, x = 5.5
Stating the Solution Therefore, the solution to the system of equations is x = 2 11 and y = 2 . This can be written as the ordered pair ( 2 11 , 2 ) or ( 5.5 , 2 ) .
Final Answer Since none of the provided options match the correct solution, the answer is none of these.
Examples
Systems of equations are used in various real-life scenarios, 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 find the number of units they need to sell to cover their costs and start making a profit. Understanding how to solve these systems allows for informed decision-making in many practical situations.
We solved the first equation for y and found y = 2 . However, substituting this value into the second equation resulted in a contradiction, indicating that the system of equations has no solution. Thus, the final answer is "none of these."
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