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Chapter 4: Problem 12

Solve each system of equations by the substitution method. See Examples I through 6 $$ \left\\{\begin{array}{l} {x+3 y=-5} \\ {2 x+2 y=6} \end{array}\right. $$

Short Answer

Expert verified
The solution to the system is \(x = 7\) and \(y = -4\).

Step by step solution

01

Choose an Equation to Solve for a Variable

Select the first equation, \(x + 3y = -5\), to solve for \(x\). This is often simpler if the variable can be easily isolated.
02

Solve for the Chosen Variable

Rearrange the equation \(x + 3y = -5\) to express \(x\) in terms of \(y\). Subtract \(3y\) from both sides to get \(x = -5 - 3y\).
03

Substitute the Expression into the Second Equation

Take the expression for \(x\) from Step 2, \(x = -5 - 3y\), and substitute it into the second equation, \(2x + 2y = 6\). This yields \(2(-5 - 3y) + 2y = 6\).
04

Simplify and Solve for \(y\)

Distribute the \(2\) in the substituted equation: \(-10 - 6y + 2y = 6\). Combine like terms to get \(-10 - 4y = 6\). Add \(10\) to both sides to obtain \(-4y = 16\). Finally, divide by \(-4\) to solve for \(y\), yielding \(y = -4\).
05

Substitute Back to Find \(x\)

Use the value of \(y = -4\) in the expression from Step 2, \(x = -5 - 3y\). Substitute \(y\) to get \(x = -5 - 3(-4)\). Simplify to find \(x = -5 + 12\), which gives \(x = 7\).
06

Verify the Solution

Substitute \(x = 7\) and \(y = -4\) back into both original equations to ensure they hold true: \(x + 3y = -5\) becomes \(7 + 3(-4) = 7 - 12 = -5\), and \(2x + 2y = 6\) becomes \(2(7) + 2(-4) = 14 - 8 = 6\). Both equations are satisfied, confirming the solution is correct.

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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 consists of multiple equations that share common variables. These systems are fundamental in algebra because they allow us to find values of variables that satisfy all equations in the system simultaneously.
When dealing with a system of equations, we aim to find the values of variables like \(x\) and \(y\) that make all the given equations true at the same time. This consistency across all equations is what defines a system.
Our example problem has a simple system with two linear equations:
  • \(x + 3y = -5\)
  • \(2x + 2y = 6\)
Such systems can often be solved graphically, but algebraic methods like the substitution method provide a more precise solution.
Algebraic Solution
An algebraic solution involves using algebraic manipulations to solve equations. It's often the most reliable way to handle complex systems since it strictly follows algebra rules.
In the substitution method, we begin by manipulating one of the equations to isolate a single variable. This equation is then expressed in terms of another variable.
From the example above, the first equation is rearranged to find \(x\):
  • \(x = -5 - 3y\)
This expression is then substituted into the other equation to solve for the other variable. This step-by-step manipulation helps us focus on one variable at a time, simplifying the process.
Solving Equations
Solving equations involves finding the values of variables that make an equation true. This is a crucial skill in algebra as it underpins virtually all mathematical problem solving.
To solve the system using the substitution method, we substitute the expression for \(x\) into the second equation:
  • Substitute \(x = -5 - 3y\) into \(2x + 2y = 6\) to get:
    \(2(-5 - 3y) + 2y = 6\)
  • Simplifying, we find
    \(-10 - 6y + 2y = 6\)
  • Combine like terms and solve for \(y\):
    \(-4y = 16\)
    \(y = -4\)
After finding \(y\), substitute back to find \(x\):
  • \(x = -5 - 3(-4) = 7\)
Finally, verify the solution by plugging \(x = 7\) and \(y = -4\) back into the original equations. Each should hold true, confirming our solution.

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Most popular questions from this chapter

The perimeter of a quadrilateral (four-sided polygon) is 29 inches. The longest side is twice as long as the shortest side. The other two sides are equally long and are 2 inches longer than the shortest side. Find the length of all four sides.

Below are two tables of values for two linear equations. Using the tables, a. find a solution of the corresponding system. b. graph several ordered pairs from each table and sketch the two lines. c. Does your graph confirm the solution from part a? $$ \begin{array}{|c|c|} \hline x & {y} \\ \hline 1 & {3} \\ \hline 2 & {5} \\ \hline 3 & {7} \\ \hline 4 & {9} \\ \hline 5 & {11} \\ \hline \end{array} $$ $$ \begin{array}{|c|c|} \hline x & {y} \\ \hline 1 & {6} \\ \hline 2 & {7} \\ \hline 3 & {8} \\ \hline 4 & {9} \\ \hline 5 & {10} \\ \hline \end{array} $$

Given the cost function \(C(x)\) and the revenue function \(R(x)\), find the number of units \(x\) that must be sold to break even. See Example 6. $$ C(x)=75 x+160,000 R(x)=200 x $$

During the 2006 regular ML.B season, Ryan Howard of the Philadelphia Phillies hit the most home runs of any player in the major leagues. Over the course of the season, he hit 4 more home runs than David Ortiz of the Boston Red Sox. Together, these batting giants hit 112 home runs. How many home runs did each player hit? (Source: Major League Baseball)

Without graphing, decide. See Examples 7 and \(8 .\) a. Are the graphs of the equations identical lines, parallel lines, or lines intersecting at a single point? b. How many solutions does the system have? $$ \left\\{\begin{array}{l} {x=5} \\ {y=-2} \end{array}\right. $$
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