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Chapter 3: Problem 81

Solve. $$ \begin{aligned} &\frac{x+y}{2}-\frac{x-y}{5}=1\\\ &\frac{x-y}{2}+\frac{x+y}{6}=-2 \end{aligned} $$

Short Answer

Expert verified
\(x = -\frac{32}{17}\) and \(y = \frac{266}{119}\)

Step by step solution

01

Simplify the Equations

Start by simplifying each equation to eliminate fractions. The first equation is \(\frac{x+y}{2}-\frac{x-y}{5}=1\) Multiply every term by 10 to get: \(5(x+y)-2(x-y)=10\)
02

Distribute and Simplify

Now distribute and simplify the equation. The first equation becomes: \(5x+5y-2x+2y=10\) Combine like terms to get: \(3x + 7y = 10\)
03

Simplify the Second Equation

For the second equation, \(\frac{x-y}{2} + \frac{x+y}{6}=-2\), Multiply every term by 6 to get: \(3(x-y) + (x+y) = -12\)
04

Distribute and Simplify Second Equation

Distribute and then simplify: \(3x-3y + x + y = -12\) Combine like terms: \(4x - 2y = -12\)
05

Solve the System of Equations

We now have the two equations: \(3x + 7y = 10\) \(4x - 2y = -12\) Let's use the method of elimination. Multiply the first equation by 2 and the second equation by 7: \(6x + 14y = 20\) \(28x - 14y = -84\) Add the two equations together: \(34x = -64\)
06

Solve for x

Divide by 34 to find \(x\): \(x = -\frac{64}{34}\) Simplify: \(x = -\frac{32}{17}\)
07

Substitute x and Solve for y

Substitute \(x\) into the first simplified equation: \(3(-\frac{32}{17}) + 7y = 10\) Simplify: \(-\frac{96}{17} + 7y = 10\) Combine \(\frac{-96}{17}\) into a single fraction with same denominator: \(7y = \frac{170}{17} + \frac{96}{17}\) \(7y = \frac{266}{17}\) Solve for \(y\): \(y = \frac{266}{(17 \cdot 7)} = \frac{266}{119}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

System of Equations
A system of equations is a set of two or more equations with the same variables. The main goal when solving a system of equations is to find the values of the variables that satisfy all equations simultaneously. In our example, we have two equations involving the variables \(x\) and \(y\):
  • \frac{x+y}{2}-\frac{x-y}{5}=1
  • \frac{x-y}{2}+\frac{x+y}{6}=-2
Such systems can be solved using various methods, like substitution, graphing, and elimination. We will focus on the elimination method.
Elimination Method
Among the methods to solve a system of equations, the elimination method is very effective when you want to eliminate one variable by combining the equations. Here's how it works:
  • First, you multiply each equation by a number that allows one of the variables to be eliminated when the two equations are added or subtracted.
  • In our example, we first simplified the original equations by eliminating fractions and then used multiplication to create coefficients that cancel each other out when adding the equations.
  • We transformed our equations to: \(3x + 7y = 10\) and \(4x - 2y = -12\). By multiplying the first equation by 2 and the second by 7, we set up the system to add up and eliminate \(y\): \(6x + 14y = 20\) and \(28x - 14y = -84\).
  • Adding these, we get \(34x = -64\), eliminating \(y\) and letting us solve directly for \(x\).
Once \(x\) is found, you substitute it back into one of the original equations to find \(y\). This systematic elimination makes solving the system more manageable.
Fraction Simplification
Simplifying fractions is a key step in solving algebraic equations, especially when they appear in a system of equations. Fraction simplification involves finding an equivalent equation without fractions by multiplying all terms by the common denominator of the fractions involved.
In our equations:
  • \frac{x+y}{2}-\frac{x-y}{5}=1
  • \frac{x-y}{2}+\frac{x+y}{6}=-2
We simplified by multiplying each term by the least common multiple (LCM) of the denominators (10 for the first and 6 for the second equation), removing the fractions, resulting in:
  • 5(x + y) - 2(x - y) = 10
  • 3(x - y) + x + y = -12
This step converts the equations into a form easier to handle: \(5x + 5y - 2x + 2y = 10\) and \(3x - 3y + x + y = -12\).
Distribution and Combination
Distribution involves multiplying each term inside a parenthesis by a factor outside the parenthesis. This is fundamental in algebra to simplify expressions. In our equations, after eliminating fractions, we distributed the terms as follows:
  • \(5(x+y) - 2(x-y)=10\) becomes \(5x + 5y - 2x + 2y = 10\)
  • \(3(x - y) + (x + y) = -12\) becomes \(3x - 3y + x + y = -12\)
Combining like terms is another crucial step where you add or subtract coefficients of the same variables to simplify further. After distribution, we combined terms:
  • \(3x + 7y = 10\) by combining \(5x - 2x\) and \(5y + 2y\)
  • \(4x - 2y = -12\) by combining \(3x + x\) and \(-3y + y\)
This simplification makes the system easier to solve through the elimination method or any other method you prefer.

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