Algebra I Chapter 7 Notes: Solving Systems of Equations and Inequalities
Section 7.1 and 7.4: Solving Systems of Equations by Graphing
System of equations: two or more equations with the same variables.
Graphing: solve both systems for y. Graph the lines on the same plane and use the point of
intersection to estimate a solution. Check your solution by back substituting.
Only 3 possible outcomes exist when solving a system of equations by graphing:
The lines can be:
1. intersecting: Has 1 solution so we call it a consistent, independent system
2. parallel: Has NO SOLUTION so we call it an inconsistent system
3. same line: Has an infinite number solutions so we call it consistent dependent. This happens
when a line is graphed on top of itself.
Classifying Systems of Equations
# of Solutions
Intersecting lines
Parallel lines
Yes
No
Consistent At least 1 solution
No
Yes
Inconsistent No solutions
Exactly
1
solution
Yes
No
Independent
No
No
Dependent Infinite # of solutions
Consistent and Independent
Intersecting lines: 1 solution
Same line
Yes
No
No
Yes
Consistent and Dependent
Same line: infinitely many solutions
Inconsistent
Parallel Lines: No solution
Consistent: The system has at least one solution (could have more than one solution), intersects at
one point or same line.
Independent: The system has exactly one solution, intersects at one point.
Inconsistent: if it has no solutions, parallel lines.
Dependent: if it has an infinite number of solutions, same line.
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Algebra I Chapter 7 Notes: Solving Systems of Equations and Inequalities
Steps for Solving Systems of Equations:
1. Solve both equations for y.
2. Put both equations into the calculator.
3. Graph.
4. Determine the characteristics of the graph(s).
There are 3 ways to solve systems of equations:
1. Graphing
2. Substitution
3. Elimination
Other ways to solve systems of equations discussed next year in Algebra II:
4. Determinants
5. Matrices
Section 7.2 – 7.3: Solving Systems of Equations Algebraically
Substitution method: one equation is solved for one variable in terms of the other. Then this
expression is substituted for that variable in the other equation.
Elimination method: eliminate one of the variables by adding or subtracting the equations.
Sometimes it will be necessary to multiply one of both equations by something before they can be
added together. Once you get one variable go back to one of the original equations to find the other
variable.
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Algebra I Chapter 7 Notes: Solving Systems of Equations and Inequalities
Section 7.5: Solving Systems of Inequalities by Graphing
System of Inequalities: find the ordered pairs that satisfy all of the inequalities in the system.
Graph the inequalities on the coordinate plane. If the inequality has a ,  the boundary line will be
solid. If the inequality has a <, > the boundary line will be dashed. To determine which side of the
boundary line, test a point that does not fall on the boundary line in the original inequality to see if
it makes the inequality true. If it is true you shade the side with your test point. If it is false you
shade the other side. The easiest point to test is (0, 0). The solution is represented by the
intersection of the graph. This region is called the feasible region.
To graph inequalities on your calculator: Enter the inequality into y1, then scroll over to the very
left of y1. Keep hitting enter until you have selected the side you want shaded. If the inequality is >
you hit the blinking triangle in the upper right corner; highlight the lower left corner for <. The
calculator will not distinguish between a solid or a dashed line.
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Systems of linear equations and inequalities can be used to model real-world situations in
which many conditions must be met.
For example, hurricanes are classified using inequalities that involve wind speed and storm
surge.
A school system wants to hire a combination of teacher and teacher aides. Given a set of
constraints, what combination of teachers and aides would be least costly?
A pet store sells dogs and cats. Given a set of constraints, what combination of dogs and
cats should they sell to maximize their profit?
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