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Lesson 8 Homework Practice Solve Systems Of Equations Algebraically Key

A system of a linear equation comprises two or more equations and one seeks a common solution to the equations. In a system of linear equations, each equation corresponds with a straight line corresponds and one seeks out the point where the two lines intersect.


Example

Solve the following system of linear equations:

$$\left\{\begin{matrix} y=2x+4\\ y=3x+2\\ \end{matrix}\right.$$

Since we are seeking out the point of intersection, we may graph the equations:

We see here that the lines intersect each other at the point x = 2, y = 8. This is our solution and we may refer to it as a graphic solution to the task.

But how does one reach a solution if the lines never intersect? One cannot, the system of equations have no solution.

One may also arrive at the correct answer with the help of the elimination method (also called the addition method or the linear combination method) or the substitution method.

When using the substitution method we use the fact that if two expressions y and x are of equal value x=y, then x may replace y or vice versa in another expression without changing the value of the expression.


Example

Solve the systems of equations using the substitution method

$$\left\{\begin{matrix} y=2x+4\\ y=3x+2\\ \end{matrix}\right.$$

We substitute the y in the top equation with the expression for the second equation:

$$\begin{array}{lcl} 2x+4 & = & 3x+2\\ 4-2 & = & 3x-2x\\ 2 & = & x\\ \end{array}$$

To determine the y-value, we may proceed by inserting our x-value in any of the equations. We select the first equation:

$$y=2x+4$$

We plug in x=2 and get

$$y=2\cdot 2+4=8$$

We have thus arrived at precisely the same answer as in the graphic solution.

The elimination method requires us to add or subtract the equations in order to eliminate either x or y, often one may not proceed with the addition directly without first multiplying either the first or second equation by some value.


Example

$$2x-2y=8$$

$$x+y=1$$

We now wish to add the two equations but it will not result in either x or y being eliminated. Therefore we must multiply the second equation by 2 on both sides and get:

$$2x-2y=8$$

$$2x+2y=2$$

Now we attempt to add our system of equations. We commence with the x-terms on the left, and the y-terms thereafter and finally with the numbers on the right side:

$$(2x+2x)+(-2y+2y)=8+2$$

The y-terms have now been eliminated and we now have an equation with only one variable:

$$4x=10$$

$$x=\frac{10}{4}=2.5$$

Thereafter, in order to determine the y-value we insert x=2.5 in one of the equations. We select the first:

$$\begin{array}{lcl} 2\cdot 2.5-2y & = & 8\\ 5-8 & = & 2y\\ -3 & = & 2y\\ \frac{-3}{2} & = & y\\ y & = & -1.5\\ \end{array}$$


Video lesson

Solve the system of equations:

$$\left\{\begin{matrix} 2x-4y=0\\ -4x+4y=-4 \end{matrix}\right.$$

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Chapter 1

Chapter 1

1-1Expressions and FormulasSkills Practicep.1
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1-2Properties of Real NumbersSkills Practicep.3
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1-3Solving EquationsSkills Practicep.5
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1-4Solving Absolute Value EquationsSkills Practicep.7
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1-5Solving InequalitiesSkills Practicep.9
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1-6Solving Compound and Absolute Value InequalitiesSkills Practicep.11
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Chapter 2

Chapter 2

2-1Relations and FunctionsSkills Practicep.13
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2-2Linear Relations and FunctionsSkills Practicep.15
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2-3Rate of Change and SlopeSkills Practicep.17
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2-4Writing Linear EquationsSkills Practicep.19
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2-5Scatter Plots and Lines of RegressionsSkills Practicep.21
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2-6Special FunctionsSkills Practicep.23
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2-7Parent Functions and TransformationsSkills Practicep.25
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2-8Graphing Linear Absolute Value InequalitiesSkills Practicep.27
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Chapter 3

Chapter 3

3-1Solving Systems of Equations by GraphingSkills Practicep.29
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3-2Solving Systems of Equations AlgebraicallySkills Practicep.31
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3-3Solving Systems of Inequalities by GraphingSkills Practicep.33
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3-4Optimization with Linear ProgrammingSkills Practicep.35
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3-5Systems of Equations in Three VariablesSkills Practicep.37
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Chapter 4

Chapter 4

4-1Introduction to MatricesSkills Practicep.39
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4-2Operations with MatricesSkills Practicep.41
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4-3Multiplying MatricesSkills Practicep.43
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4-4Transformations with MatricesSkills Practicep.45
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4-5Determinants and Cramer's RuleSkills Practicep.47
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4-6Inverse Matrices and Systems of EquationsSkills Practicep.49
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Chapter 5

Chapter 5

5-1Graphing Quadratic EquationsSkills Practicep.51
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5-2Solving Quadratic Equations by GraphingSkills Practicep.53
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5-3Solving Quadratic Equations by FactoringSkills Practicep.55
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5-4Complex NumbersSkills Practicep.57
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5-5Completing the SquareSkills Practicep.59
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5-6The Quadratic Formula and the DiscriminantSkills Practicep.61
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5-7Transformations with Quadratic FunctionsSkills Practicep.63
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5-8Quadratic InequalitiesSkills Practicep.65
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Chapter 6

Chapter 6

6-1Operations with PolynomialsSkills Practicep.67
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6-2Dividing PolynomialsSkills Practicep.69
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6-3Polynomial FunctionsSkills Practicep.71
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6-4Analyzing Graphs of Polynomial FunctionsSkills Practicep.73
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6-5Solving Polynomial EquationsSkills Practicep.75
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6-6The Remainder and Factor TheoremsSkills Practicep.77
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6-7Roots and ZerosSkills Practicep.79
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6-8Rational Zero TheoremSkills Practicep.81
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Chapter 7

Chapter 7

7-1Operations on FunctionsSkills Practicep.83
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7-2Inverse Functions and RelationsSkills Practicep.85
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7-3Square Root Functions and InequalitiesSkills Practicep.87
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7-4nth RootsSkills Practicep.89
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7-5Operations with Radical ExpressionsSkills Practicep.91
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7-6Rational ExponentsSkills Practicep.93
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7-7Solving Radical Equations and InequalitiesSkills Practicep.95
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Chapter 8

Chapter 8

8-1Graphing Exponential FunctionsSkills Practicep.97
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8-2Solving Exponential Equations and InequalitiesSkills Practicep.99
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8-3Logarithms and Logarithmic FunctionsSkills Practicep.101
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8-4Solving Logarithmic Equations and InequalitiesSkills Practicep.103
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8-5Properties of LogarithmsSkills Practicep.105
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8-6Common LogarithmsSkills Practicep.107
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8-7Base e and Natural LogarithmsSkills Practicep.109
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8-8Using Exponential and Logarithmic FunctionsSkills Practicep.111
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Chapter 9

Chapter 9

9-1Multiplying and Dividing Rational ExpressionsSkills Practicep.113
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9-2Adding and Subtracting Rational ExpressionsSkills Practicep.115
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9-3Graphing Reciprocal FunctionsSkills Practicep.117
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9-4Graphing Rational FunctionsSkills Practicep.119
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9-5Variation FunctionsSkills Practicep.121
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9-6Solving Rational Equations and InequalitiesSkills Practicep.123
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Chapter 10

Chapter 10

10-1Midpoint and Distance FormulasSkills Practicep.125
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10-2ParabolasSkills Practicep.127
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10-3CirclesSkills Practicep.129
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10-4EllipsesSkills Practicep.131
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10-5HyperbolasSkills Practicep.133
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10-6Identifying Conic SectionsSkills Practicep.135
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10-7Solving Quadratic SystemsSkills Practicep.137
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Chapter 11

Chapter 11

11-1Sequences as FunctionsSkills Practicep.139
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11-2Arithmetic Sequences and SeriesSkills Practicep.141
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11-3Geometric Sequences and SeriesSkills Practicep.143
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11-4Infinite Geometric SeriesSkills Practicep.145
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11-5Recursion and IterationSkills Practicep.147
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11-6The Binomial TheoremSkills Practicep.149
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11-7Proof by Mathematical InductionSkills Practicep.151
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Chapter 12

Chapter 12

12-1Experiments, Surveys, and Observational StudiesSkills Practicep.153
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12-2Statistical AnalysisSkills Practicep.155
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12-3Conditional ProbabilitySkills Practicep.157
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12-4Probability DistributionsSkills Practicep.159
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12-5The Normal DistributionSkills Practicep.161
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12-6Hypothesis TestingSkills Practicep.163
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12-7Binomial DistributionsSkills Practicep.165
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Chapter 13

Chapter 13

13-1Trigonometric Functions in Right TrianglesSkills Practicep.167
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13-2Angles and Angle MeasureSkills Practicep.169
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13-3Trigonometric Functions of General AnglesSkills Practicep.171
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13-4Law of SinesSkills Practicep.173
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13-5Law of CosinesSkills Practicep.175
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13-6Circular FunctionsSkills Practicep.177
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13-7Graphing Trigonometric FunctionsSkills Practicep.179
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13-8Translations of Trigonometric GraphsSkills Practicep.181
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13-9Inverse Trigonometric FunctionsSkills Practicep.183
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Chapter 14

Chapter 14

14-1Trigonometric IdentitiesSkills Practicep.185
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14-2Verifying Trigonometric IdentitiesSkills Practicep.187
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14-3Sum and Difference of Angles FormulasSkills Practicep.189
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14-4Double-Angle and Half-Angle FormulasSkills Practicep.191
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14-5Solving Trigonometric EquationsSkills Practicep.193
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