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Chapter 8: Problem 17

Solve each system by the elimination method. $$ \begin{aligned} &3 x+3 y=33\\\ &5 x-2 y=27 \end{aligned} $$

Short Answer

Expert verified
The solution is \((7, 4)\).

Step by step solution

01

- Write down the equations

First, write down both equations from the system: 1. \(3x + 3y = 33\) 2. \(5x - 2y = 27\)
02

- Multiply one or both equations to align coefficients

To eliminate a variable, align the coefficients of one of the variables. Multiply the first equation by 2 and the second equation by 3 so the coefficients of y will be equal in magnitude: Equation 1: \(2(3x + 3y) = 2 \cdot 33\) which gives \(6x + 6y = 66\) Equation 2: \(3(5x - 2y) = 3 \cdot 27\) which gives \(15x - 6y = 81\)
03

- Add or subtract the equations to eliminate y

Add the two equations to eliminate the y variable: \[(6x + 6y) + (15x - 6y) = 66 + 81\] This simplifies to: \(21x = 147\)
04

- Solve for x

Solve for x by dividing both sides of the equation by 21: \(x = \frac{147}{21}\) This simplifies to: \(x = 7\)
05

- Substitute x back into one of the original equations

Substitute \(x = 7\) into the first original equation to find y: \(3(7) + 3y = 33\) Simplify to: \(21 + 3y = 33\)
06

- Solve for y

Isolate y by first subtracting 21 from both sides: \(3y = 12\) Then divide by 3: \(y = 4\)
07

- Write the solution as an ordered pair

The solution to the system of equations is \((x, y) = (7, 4)\)

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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 technique used to solve systems of linear equations. The fundamental idea is to eliminate one of the variables by adding or subtracting the equations.
This allows you to solve for the remaining variable more easily.
In order to use the elimination method, you typically follow these steps:

  • Write down both equations from the system.
  • Multiply one or both equations by a number so that the coefficients of one of the variables are opposites.
  • Add or subtract the equations to eliminate that variable.
  • Solve for the remaining variable.
  • Substitute this back into one of the original equations to find the other variable.
By following these steps, you can systematically solve for both variables in the system.
Linear Equations
A linear equation is an equation that makes a straight line when graphed.
Linear equations can usually be written in the form \(ax + by = c\), where \a\, \b\, and \c\ are constants. For example, the equations \(3x + 3y = 33\) and \(5x - 2y = 27\) are linear.

Here are some important properties of linear equations:
  • They have constants and variables.
  • The highest power of the variable is 1.
  • They can be manipulated through addition, subtraction, multiplication, and division.
Linear equations are the building blocks for understanding more complex algebraic solutions.
Algebraic Solutions
An algebraic solution refers to the process of solving equations using algebraic methods.
This typically involves isolating the variable and performing operations to find its value.
In the context of systems of linear equations, algebraic solutions usually involve:

  • Rewriting the equations to align the coefficients.
  • Using techniques like the elimination method to simplify the system.
  • Solving for one of the variables, and then substituting back to find the other.
  • Expressing the solution as an ordered pair \((x, y)\).
Understanding algebraic solutions is crucial for tackling problems involving linear equations and many other types of equations in mathematics.

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Most popular questions from this chapter

In \(2009,\) the FCI prices for Major League Baseball and the National Football League to taled 609.53 dollar. The football FCI was 215.75 more than that of baseball. What were the FCIs for these sports?

Solve system of equations by graphing. If the system is inconsistent or the equations are dependent, say so. \(4 x+y=5\) \(y=\frac{3}{2} x-6\)

During the \(2008-2009 \mathrm{NHL}\) regular season, the Boston Bruins played 82 games. Their wins and overtime losses resulted in a total of 116 points. They had 9 more losses in regulation play than overtime losses. How many wins, losses, and overtime losses did they have that year? \(\begin{array}{|c|c|c|c|c|}\hline \text { Team } & {G P} & {W} & {L} & {O T L} & {P \text { oints }} \\ {\text { Boston }} & {82} & {} & {} & {} & {116} \\\ {\text { Montreal }} & {82} & {41} & {30} & {11} & {93} \\ {\text { Buffalo }} & {82} & {41} & {32} & {9} & {91} \\ {\text { Ottawa }} & {82} & {36} & {35} & {11} & {83} \\ {\text { Toronto }} & {82} & {34} & {35} & {13} & {81} \\\ \hline\end{array}\)

Solve system of equations by graphing. If the system is inconsistent or the equations are dependent, say so. \(x-2 y=6\) \(x+2 y=2\)

Solve each problem. How many gallons each of \(25 \%\) alcohol and \(35 \%\) alcohol should be mixed to get 20 gal of \(32 \%\) alcohol? $$\begin{array}{|c|c|c|}\hline \text { Gallons } & {\text { Percent }} & {\text { Gallons of }} \\ {\text { of Solution }} & {\text { (as a decimal) }} & {\text { Pure Alcohol }} \\ {x} & {25 \%=0.25} \\ {y} & {35 \%=0.35} \\\ {\text { 20 }} & {32 \%= 0.32}\end{array}$$
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