Hartfield – College Algebra (Version 2014-3D) Unit EIGHT | Page 1
Topic 8-1: Systems of Two Linear Equations
Definition: A system of equations is a set of equations involving the same variables simultaneously.
A system of equations is said to be solved when a set of values for each variable in the system satisfies every equation in the system.
Systems of linear equations are those systems where every equation is linear. In college algebra we will want to solve systems of linear equations.
A system of nonlinear equations is a system with one or more nonlinear equation and will be covered in precalculus.
Solving a system of two linear equations can be accomplished in four ways:
1. Finding the intersection point of the graphs of each equation.
2. Substituting one equation into the other to find the value of one variable and then using that value to find the value of the second variable. 3. Adding equations together, after appropriate
multiplicative operations to one or both
equations, to find the value of one variable. Then using that value to find the value of the second variable.
4. Corresponding the coefficients of the system to a matrix and applying elementary row
operations to create an identity matrix and a solution matrix.
We will not apply matrices to solve systems in this class. Our primary approaches will be methods 2 and 3, the Substitution Method and the Elimination method, respectively, with supplemental graphing for confirmation.
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The Substitution Method requires the following set of steps:
1. Solve either equation for x or y.
2. Plug that equation in for the variable in the other equation.
3. Solve the created equation involving one variable.
4. Plug the solution found in step 3 into any equation involving two variables & solve. 5. Write your solution as an ordered pair.
The Elimination Method requires the following set of steps:
1. As necessary, multiply one or both
coefficients by a positive or negative whole number so that a column of coefficients have the same absolute value but different signs. 2. Add the equations to eliminate a variable and
create a new equation.
3. Plug the solution found in step 2 into any equation involving two variables & solve. 4. Write your solution as an ordered pair.
Ex. Solve the system of equations using the Substitution Method. 4 4 3 2 16 x y x y
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Ex. Solve the system of equations using the Elimination Method. 2 1 3 2 4 x y x y
Observe that if each system is graphed, the
solution corresponds with the intersection point of the equations in the system.
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Ex. 3 Solve the system of equations.
5 3 2 2 4 6 x y x y
Most systems of linear equations have exactly one solution. A system that has exactly one solution is classified as being consistent and independent.
A system of linear equation can have no solution if the graphs of the equations are parallel (as thus lack an intersection point). Such a system is classified as being inconsistent.
When solving a system through either method, if a contradiction is reached (such as two
different numbers supposedly being equal to each other), the system must be inconsistent.
A system of linear equations can have
infinitely-many solutions if the equations are not unique (and thus share all their points along a coincident line). Such a system is classified as being dependent.
When solving a system through either method, if an identity statement is reached (such as a number being equal to itself), the system must be dependent.
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Topic 8-2: Systems of Three Linear Equations
To solve a system of three linear equations, we
wish to rewrite it into triangular form. The following pair of systems is equivalent, that is, they have the same ordered triple for a solution, but the one on the right is in triangular form.
2 3 1 2 2 3 3 2 4 x y y x y z x y z 2 3 1 5 8 1 2 x y y y z z
The system in triangular form is easy to solve through back-substitution, thus making it a preferable form for solving the system.
The Gaussian Elimination Method for rewriting a system into triangular form requires some
combination of the following set of operations:
1. Add a nonzero multiple of one equation to another.
2. Multiply an equation by a nonzero constant. 3. Interchange the positions of two equations.
Once triangular form is reached, use
back-substitution to express the solution as an ordered triple.
Observation: If a contradiction of an identity statement is found through the process of rewriting into triangular form, then the system has no
solution or infinitely-many solutions, respectively.
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Ex. 1 Rewrite the system of equations into triangular form and then solve.
2 2 3 5 8 2 2 7 x y z x y z x y z
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Ex. 2 Rewrite the system of equations into triangular form and then solve.
0 3 6 2 5 3 x y z x y x y z
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Ex. 3 Rewrite the system of equations into triangular form and then solve.
2 3 8 5 2 4 5 1 x y z x y z x y z