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Chapter 7: Problem 46

Solve each system by elimination. $$\begin{aligned}&\frac{x+6}{5}+\frac{2 y-x}{10}=1\\\&\frac{x+2}{4}+\frac{3 y+2}{5}=-3\end{aligned}$$

Short Answer

Expert verified
The solution is \( x = 66 \) and \( y = -34 \).

Step by step solution

01

Simplify the Equations

First, simplify the fractions in each equation to make them easier to work with.Equation 1: \( \frac{x+6}{5} + \frac{2y-x}{10} = 1 \) becomes: Multiply the first term by 2 to have a common denominator: \( \frac{2(x+6)}{10} + \frac{2y-x}{10} = 1 \).Combine under a common denominator: \( \frac{2x + 12 + 2y - x}{10} = 1 \). Simplify: \( \frac{x + 2y + 12}{10} = 1 \).Equation 2: \( \frac{x+2}{4} + \frac{3y+2}{5} = -3 \) becomes: Multiply the first term by 5 and the second term by 4 to have a common denominator of 20: \( \frac{5(x+2)}{20} + \frac{4(3y+2)}{20} = -3 \).Combine under a common denominator: \( \frac{5x + 10 + 12y + 8}{20} = -3 \). Simplify: \( \frac{5x + 12y + 18}{20} = -3 \).
02

Eliminate the Fractions

Clear the fractions by multiplying each equation through by the least common multiple (LCM) of the denominators.Equation 1: The LCM of 10 is 10. Multiply through to get:\( x + 2y + 12 = 10 \).Equation 2: The LCM of 20 is 20. Multiply through to get:\( 5x + 12y + 18 = -60 \).
03

Align the System

Write each equation in standard form where similar terms are aligned.Equation 1: \( x + 2y = -2 \) (we moved 12 to the other side by subtracting it from both sides).Equation 2: \( 5x + 12y = -78 \) (we subtracted 18 from both sides).
04

Eliminate One Variable

Choose a variable to eliminate using the elimination method. Here, we will eliminate \( x \).Multiply the first equation by 5 (coefficient of \( x \) in the second equation) to align \( x \):Equation 1 becomes: \( 5x + 10y = -10 \).Now subtract this result from Equation 2:\( (5x + 12y = -78) - (5x + 10y = -10) \).The \( x \) terms cancel out, resulting in:\( 2y = -68 \).
05

Solve for Remaining Variable

Solve for \( y \):Divide both sides by 2:\( y = -34 \).
06

Solve for the Other Variable

Substitute \( y = -34 \) back into one of the original simplified equations to solve for \( x \). We'll use Equation 1: \( x + 2y = -2 \).Substitute \( y = -34 \):\( x + 2(-34) = -2 \).Simplify:\( x - 68 = -2 \).Add 68 to both sides:\( x = 66 \).
07

Solution Check

Verify the solution by substituting \( x = 66 \) and \( y = -34 \) back into the original equations.For Equation 1: \( \frac{66+6}{5} + \frac{2(-34)-66}{10} = 1 \), this simplifies correctly.For Equation 2: \( \frac{66+2}{4} + \frac{3(-34)+2}{5} = -3 \), this simplifies correctly.Both equations satisfy the solutions, confirming they are correct.

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Key Concepts

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

Elimination Method
The elimination method is a fundamental technique used to solve systems of equations, particularly linear equations. This method involves eliminating one of the variables in a system so you can solve for the other variable easily.
  • **How it Works**: Start by aligning the equations. You may need to multiply one or both equations by a certain number to ensure one variable has the same coefficient in both equations.

  • **Reduction**: Subtract or add the equations to cancel out one variable. This leaves you with a single equation containing only one variable, which is much easier to solve.
  • **Example**: Suppose you are given \( x + 2y = -2 \) and \( 5x + 12y = -78 \). To eliminate \( x \), multiply the first equation by 5, resulting in \( 5x + 10y = -10 \). Subtract this new equation from the second to get \( 2y = -68 \).
Once you have a solution for one variable, substitute it back into one of the original equations to find the other variable. This method gives you a precise and effective tool for solving systems of equations.
Solving Fractions
Solving equations with fractions can seem a bit challenging, but it becomes straightforward when you apply the right techniques. A common approach is to eliminate the fractions by finding a common denominator.
  • **Identifying the Common Denominator**: Look at all the denominators in your equations. Find the least common multiple (LCM) of these denominators.

  • **Clearing Fractions**: Multiply each term of the equation by the LCM. This step transforms the equation into one without fractions, simplifying it significantly.

  • **Example**: For the system \( \frac{x+6}{5} + \frac{2y-x}{10} = 1 \) and \( \frac{x+2}{4} + \frac{3y+2}{5} = -3 \), identify the LCMs (10 and 20). Multiply through to remove fractions, simplifying your work.
Once fractions are cleared, solving these now simplified equations using other methods like elimination becomes easier. The process of solving with fractions involves transforming the equations into a form that's more straightforward to deal with.
Linear Equations
Linear equations are equations where each term is either a constant or the product of a constant and a single variable. They form straight lines when graphed.
  • **Standard Form**: These equations are commonly expressed as \( ax + by = c \). Each variable is to the first power, and there are no products of variables.
  • **Graphing**: When graphed on a coordinate plane, linear equations represent straight lines. The intersection point of two lines is the solution of the system represented by those equations.

  • **Example in Systems**: In the system \( x + 2y = -2 \) and \( 5x + 12y = -78 \), we see typical linear equations. Methods such as elimination or substitution are used to find where these lines intersect, revealing the solution.

Understanding the structure and properties of linear equations can aid in effectively employing methods, like the elimination method, to solve systems of such equations. Linear equations are not only foundational in algebra but also in various real-world applications.

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Most popular questions from this chapter

Solve each problem..Inquiries about Displayed Products \(\quad\) A wholesaler of party goods wishes to display her products at a convention of social secretaries in such a way that she gets the maximum number of inquiries about her whistles and hats. Her booth at the convention has 12 square meters of floor space to be used for display purposes. A display unit for hats requires 2 square meters, and one for whistles requires 4 square meters. Experience tells the wholesaler that she should never have more than a total of 5 units of whistles and hats on display at one time. If she receives three inquiries for each unit of hats and two inquiries for each unit of whistles on display, how many of each should she display in order to get the maximum number of inquiries? What is that maximum number?

Graph the solution set of each system of inequalities by hand. $$\begin{aligned}&y \geq(x-2)^{2}+3\\\&y \leq-(x-1)^{2}+6\end{aligned}$$

Use the shading capabilities of a graphing calculator to graph each inequality or system of inequalities. $$\begin{aligned}&y \geq 2^{x}\\\&y \leq 8\end{aligned}$$

Graph the solution set of each system of inequalities by hand. $$\begin{array}{c}x+y \geq 0 \\\2 x-y \geq 3\end{array}$$

Graph the solution set of each system of inequalities by hand. $$\begin{aligned}&y \geq \frac{1}{x+2}\\\&y \geq x-2\end{aligned}$$
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