If x+2y=3 and 2x-y=-2, find the value of (3y-x)-2(3y-x)-5
Understand the Problem
The question presents a system of two linear equations with two variables (x and y). It asks us to first solve for x and y, then substitute those values into the expression (3y-x)-2(3y-x)-5 and simplify to find the final numerical value.
Answer
-8
Answer for screen readers
-8
Steps to Solve
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Solve the system of equations for x and y We have the following system of equations: $$ x + y = 5 $$ $$ x - y = 1 $$ We can solve this system using the elimination or substitution method. Here, we'll use the elimination method. Add the two equations to eliminate $y$: $$ (x + y) + (x - y) = 5 + 1 $$ $$ 2x = 6 $$ Divide both sides by 2: $$ x = 3 $$
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Substitute the value of x into one of the original equations to solve for y Substitute $x = 3$ into the first equation: $$ 3 + y = 5 $$ Subtract 3 from both sides: $$ y = 2 $$
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Simplify the expression The expression we want to simplify is $(3y - x) - 2(3y - x) - 5$. Let's first simplify the algebraic expression before substituting the values of $x$ and $y$. We can treat $(3y - x)$ as a single term and combine like terms. $$ (3y - x) - 2(3y - x) - 5 = (1 - 2)(3y - x) - 5 = -1(3y - x) - 5 = -(3y - x) - 5 = -3y + x - 5 $$
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Substitute the values of x and y into the simplified expression Now, substitute $x = 3$ and $y = 2$ into the simplified expression $-3y + x - 5$: $$ -3(2) + 3 - 5 = -6 + 3 - 5 = -3 - 5 = -8 $$
-8
More Information
The solution to the system of equations is $x = 3$ and $y = 2$. Substituting these values into the expression $(3y-x)-2(3y-x)-5$ and simplifying, we get the final answer of $-8$.
Tips
A common mistake is to forget to distribute the $-2$ in the expression. It's also easy to make a mistake when combining the terms after the distribution or when substituting the values of $x$ and $y$.
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