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Graphical & Algebraic Solutions for Linear Systems of Equations in Two Variables - Prof. Z, Exams of Algebra

An in-depth review of chapter 4 from math 102, focusing on systems of linear equations in two variables. Topics covered include graphical solutions, elimination method, and substitution method for solving linear systems, as well as applications of these concepts to word problems involving coins, mixtures, and interest. Students should understand the meaning of a linear system of equations, how to graph lines, and how to use elimination and substitution methods to find the unique solution or determine if there is no solution or an infinite number of solutions.

Typology: Exams

Pre 2010

Uploaded on 08/17/2009

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Test 3 Review (Ch. 4) Math 102
Ch. 4: Systems of linear equations in two variables.
4.1: Introduction to linear systems/graphical solutions.
1. You should know what a linear system of equations means: we are looking for the unique
ordered pair that solves both equations.
2. Since each equation in a linear system represents a line, this means that we want the unique
point sitting on both lines. Thus, we find a graphical solution to a system by graphing each
equation in the x-y plane and looking for the intersection.
3. Special cases: if two lines are parallel, then there is no intersection (so there is no solution to
the system). If the two equations in the system are equivalent (so that each one actually
represents the same line), then we say “the lines coincide” and there are an infinite number of
solutions. You can also say “any point on the line”.
Note: you should be able to quickly graph both lines in the system by solving for y and using
slope-intercept form.
4.2: Elimination method for solving linear systems.
1. Elimination is based on the fact that we can “add equations”, meaning that we can add the left
sides of two equations and the right sides of the same equations to generate a new equation.
2. If we can eliminate a variable by adding two equations, we are left with a new equation that
has only one variable. This allows us to solve for that variable (either x or y). Once we have the
x or y coordinate of the solution point, we can get the other coordinate by substituting that value
into either of the original equations.
3. We usually have to “rig” one or both equations in order for a variable to be eliminated when
we add the equations. We do this by multiplying both sides of one (or both) equations by the
proper constant(s) so that x or y is eliminated when we add the equations.
4. Note that systems involving fractions or decimals can be simplified by multiplying both sides
of the offending equation by the least common denominator.
5. If an answer like 0=2 results from elimination, it means there is no solution to the system (the
lines are parallel). If an answer like 1=1 results, the lines coincide and there are an infinite
number of solutions.
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Test 3 Review (Ch. 4) Math 102

Ch. 4: Systems of linear equations in two variables.

4.1: Introduction to linear systems/graphical solutions.

  1. You should know what a linear system of equations means : we are looking for the unique ordered pair that solves both equations.
  2. Since each equation in a linear system represents a line, this means that we want the unique point sitting on both lines. Thus, we find a graphical solution to a system by graphing each equation in the x-y plane and looking for the intersection.
  3. Special cases: if two lines are parallel, then there is no intersection (so there is no solution to the system). If the two equations in the system are equivalent (so that each one actually represents the same line), then we say “the lines coincide” and there are an infinite number of solutions. You can also say “any point on the line”.

Note: you should be able to quickly graph both lines in the system by solving for y and using slope-intercept form.

4.2: Elimination method for solving linear systems.

  1. Elimination is based on the fact that we can “add equations”, meaning that we can add the left sides of two equations and the right sides of the same equations to generate a new equation.
  2. If we can eliminate a variable by adding two equations, we are left with a new equation that has only one variable. This allows us to solve for that variable (either x or y). Once we have the x or y coordinate of the solution point, we can get the other coordinate by substituting that value into either of the original equations.
  3. We usually have to “rig” one or both equations in order for a variable to be eliminated when we add the equations. We do this by multiplying both sides of one (or both) equations by the proper constant(s) so that x or y is eliminated when we add the equations.
  4. Note that systems involving fractions or decimals can be simplified by multiplying both sides of the offending equation by the least common denominator.
  5. If an answer like 0=2 results from elimination, it means there is no solution to the system (the lines are parallel). If an answer like 1=1 results, the lines coincide and there are an infinite number of solutions.

4.3: Substitution method for solving linear systems.

  1. We perform substitution by solving one equation in the system for one of the variables (the choice is up to you – there is usually an easy way and a hard way).
  2. Once we have solved for one variable in terms of the other, we plug that relationship into the other equation to obtain a linear equation with only one variable – this can be solved using the techniques of chapter 2.
  3. Now that we have the value for one variable, we substitute into either equation to obtain the value of the other variable.
  4. The same interpretations apply for answers like “0=2” (no solution) and “1=1” (lines coincide).

4.4: Applications of linear systems.

  1. The usual word-problem techniques apply: organize the information in a sensible way, give variable names to the unknowns, and use the given information to write a system of equations relating the variables. This time, every problem should produce a system of equations.
  2. Note on coin problems: one equation will be for the total number of coins. The other equation will represent the total value of all the coins.
  3. Note on mixture problems: one equation will represent the amount of pure “stuff” (alcohol for example). The other equation will represent the total amount of stuff (alcohol solution for example).
  4. Note on interest problems: to compute the amount of interest, you multiply the amount invested by the interest rate. One equation will be for the total amount of money, while the other is for the total amount of interest.