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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

 1-1 Expressions and Formulas Skills Practice p.1 Practice p.2 1-2 Properties of Real Numbers Skills Practice p.3 Practice p.4 1-3 Solving Equations Skills Practice p.5 Practice p.6 1-4 Solving Absolute Value Equations Skills Practice p.7 Practice p.8 1-5 Solving Inequalities Skills Practice p.9 Practice p.10 1-6 Solving Compound and Absolute Value Inequalities Skills Practice p.11 Practice p.12

### Chapter 2

Chapter 2

 2-1 Relations and Functions Skills Practice p.13 Practice p.14 2-2 Linear Relations and Functions Skills Practice p.15 Practice p.16 2-3 Rate of Change and Slope Skills Practice p.17 Practice p.18 2-4 Writing Linear Equations Skills Practice p.19 Practice p.20 2-5 Scatter Plots and Lines of Regressions Skills Practice p.21 Practice p.22 2-6 Special Functions Skills Practice p.23 Practice p.24 2-7 Parent Functions and Transformations Skills Practice p.25 Practice p.26 2-8 Graphing Linear Absolute Value Inequalities Skills Practice p.27 Practice p.28

### Chapter 3

Chapter 3

 3-1 Solving Systems of Equations by Graphing Skills Practice p.29 Practice p.30 3-2 Solving Systems of Equations Algebraically Skills Practice p.31 Practice p.32 3-3 Solving Systems of Inequalities by Graphing Skills Practice p.33 Practice p.34 3-4 Optimization with Linear Programming Skills Practice p.35 Practice p.36 3-5 Systems of Equations in Three Variables Skills Practice p.37 Practice p.38

### Chapter 4

Chapter 4

 4-1 Introduction to Matrices Skills Practice p.39 Practice p.40 4-2 Operations with Matrices Skills Practice p.41 Practice p.42 4-3 Multiplying Matrices Skills Practice p.43 Practice p.44 4-4 Transformations with Matrices Skills Practice p.45 Practice p.46 4-5 Determinants and Cramer's Rule Skills Practice p.47 Practice p.48 4-6 Inverse Matrices and Systems of Equations Skills Practice p.49 Practice p.50

### Chapter 5

Chapter 5

 5-1 Graphing Quadratic Equations Skills Practice p.51 Practice p.52 5-2 Solving Quadratic Equations by Graphing Skills Practice p.53 Practice p.54 5-3 Solving Quadratic Equations by Factoring Skills Practice p.55 Practice p.56 5-4 Complex Numbers Skills Practice p.57 Practice p.58 5-5 Completing the Square Skills Practice p.59 Practice p.60 5-6 The Quadratic Formula and the Discriminant Skills Practice p.61 Practice p.62 5-7 Transformations with Quadratic Functions Skills Practice p.63 Practice p.64 5-8 Quadratic Inequalities Skills Practice p.65 Practice p.66

### Chapter 6

Chapter 6

 6-1 Operations with Polynomials Skills Practice p.67 Practice p.68 6-2 Dividing Polynomials Skills Practice p.69 Practice p.70 6-3 Polynomial Functions Skills Practice p.71 Practice p.72 6-4 Analyzing Graphs of Polynomial Functions Skills Practice p.73 Practice p.74 6-5 Solving Polynomial Equations Skills Practice p.75 Practice p.76 6-6 The Remainder and Factor Theorems Skills Practice p.77 Practice p.78 6-7 Roots and Zeros Skills Practice p.79 Practice p.80 6-8 Rational Zero Theorem Skills Practice p.81 Practice p.82

### Chapter 7

Chapter 7

 7-1 Operations on Functions Skills Practice p.83 Practice p.84 7-2 Inverse Functions and Relations Skills Practice p.85 Practice p.86 7-3 Square Root Functions and Inequalities Skills Practice p.87 Practice p.88 7-4 nth Roots Skills Practice p.89 Practice p.90 7-5 Operations with Radical Expressions Skills Practice p.91 Practice p.92 7-6 Rational Exponents Skills Practice p.93 Practice p.94 7-7 Solving Radical Equations and Inequalities Skills Practice p.95 Practice p.96

### Chapter 8

Chapter 8

 8-1 Graphing Exponential Functions Skills Practice p.97 Practice p.98 8-2 Solving Exponential Equations and Inequalities Skills Practice p.99 Practice p.100 8-3 Logarithms and Logarithmic Functions Skills Practice p.101 Practice p.102 8-4 Solving Logarithmic Equations and Inequalities Skills Practice p.103 Practice p.104 8-5 Properties of Logarithms Skills Practice p.105 Practice p.106 8-6 Common Logarithms Skills Practice p.107 Practice p.108 8-7 Base e and Natural Logarithms Skills Practice p.109 Practice p.110 8-8 Using Exponential and Logarithmic Functions Skills Practice p.111 Practice p.112

### Chapter 9

Chapter 9

 9-1 Multiplying and Dividing Rational Expressions Skills Practice p.113 Practice p.114 9-2 Adding and Subtracting Rational Expressions Skills Practice p.115 Practice p.116 9-3 Graphing Reciprocal Functions Skills Practice p.117 Practice p.118 9-4 Graphing Rational Functions Skills Practice p.119 Practice p.120 9-5 Variation Functions Skills Practice p.121 Practice p.122 9-6 Solving Rational Equations and Inequalities Skills Practice p.123 Practice p.124

### Chapter 10

Chapter 10

 10-1 Midpoint and Distance Formulas Skills Practice p.125 Practice p.126 10-2 Parabolas Skills Practice p.127 Practice p.128 10-3 Circles Skills Practice p.129 Practice p.130 10-4 Ellipses Skills Practice p.131 Practice p.132 10-5 Hyperbolas Skills Practice p.133 Practice p.134 10-6 Identifying Conic Sections Skills Practice p.135 Practice p.136 10-7 Solving Quadratic Systems Skills Practice p.137 Practice p.138

### Chapter 11

Chapter 11

 11-1 Sequences as Functions Skills Practice p.139 Practice p.140 11-2 Arithmetic Sequences and Series Skills Practice p.141 Practice p.142 11-3 Geometric Sequences and Series Skills Practice p.143 Practice p.144 11-4 Infinite Geometric Series Skills Practice p.145 Practice p.146 11-5 Recursion and Iteration Skills Practice p.147 Practice p.148 11-6 The Binomial Theorem Skills Practice p.149 Practice p.150 11-7 Proof by Mathematical Induction Skills Practice p.151 Practice p.152

### Chapter 12

Chapter 12

 12-1 Experiments, Surveys, and Observational Studies Skills Practice p.153 Practice p.154 12-2 Statistical Analysis Skills Practice p.155 Practice p.156 12-3 Conditional Probability Skills Practice p.157 Practice p.158 12-4 Probability Distributions Skills Practice p.159 Practice p.160 12-5 The Normal Distribution Skills Practice p.161 Practice p.162 12-6 Hypothesis Testing Skills Practice p.163 Practice p.164 12-7 Binomial Distributions Skills Practice p.165 Practice p.166

### Chapter 13

Chapter 13

 13-1 Trigonometric Functions in Right Triangles Skills Practice p.167 Practice p.168 13-2 Angles and Angle Measure Skills Practice p.169 Practice p.170 13-3 Trigonometric Functions of General Angles Skills Practice p.171 Practice p.172 13-4 Law of Sines Skills Practice p.173 Practice p.174 13-5 Law of Cosines Skills Practice p.175 Practice p.176 13-6 Circular Functions Skills Practice p.177 Practice p.178 13-7 Graphing Trigonometric Functions Skills Practice p.179 Practice p.180 13-8 Translations of Trigonometric Graphs Skills Practice p.181 Practice p.182 13-9 Inverse Trigonometric Functions Skills Practice p.183 Practice p.184

### Chapter 14

Chapter 14

 14-1 Trigonometric Identities Skills Practice p.185 Practice p.186 14-2 Verifying Trigonometric Identities Skills Practice p.187 Practice p.188 14-3 Sum and Difference of Angles Formulas Skills Practice p.189 Practice p.190 14-4 Double-Angle and Half-Angle Formulas Skills Practice p.191 Practice p.192 14-5 Solving Trigonometric Equations Skills Practice p.193 Practice p.194