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Chapter 7
Chapter 7
Chapter Goals
In this chapter, students will solve systems of two linear equations &
graph the solutions to systems of linear inequalities. They will:
 Solve
linear systems by graphing, substitution, & linear
combinations
 Use
linear systems to model real life situations
 Determine
the number of solutions to a linear system
Chapter 7
Chapter Study Guide
7.1 – Graphing Linear Systems
7.2 – Solving Linear Systems by Substitution
7.3 – Solving Linear Systems by Combinations
7.4 – Linear Systems & Problem Solving
7.5 – Special Types of Linear Systems
7.6 – Systems of Linear Inequalities
Chapter 7
Key Words – Define these
 System
of linear equations
 Solution
 Point
of a linear system
of intersection
 Linear
combination
 System
of linear inequalities
 Solution
of a system of linear inequalities
7.1
7.1 Graphing Linear Systems
 Two
or more linear equations in the same variable form a system of
linear equations, or simply a linear system. Here is an example of
a linear system:
x + 2y = 5
2x + 3y = 3
 A solution of a linear system in two variables is a pair of numbers
a & b for which x = a & y = b make each equation a true statement.
 Such a solution can be written as an ordered pair (a, b) in which a &
b are the values of x & y that solve the linear system. The point (a,
b) that lies on the graph of each equation is called the point of
intersection of the graph.
7.1
Solving a Linear System Using Graph-&-Check
1. Write each equation in a form that is easy to graph
2. Graph both equations in the same coordinate plane
3. Estimate the coordinates of the point intersection
4. Check whether the coordinates give a solution by substituting
them into each equation of the original linear system.
7.2
Solving Linear Systems by Substitution
 There are several ways to solve a linear system without using
graphs. In this lesson you will study an algebraic method known as
the substitution method.
• Substitution Method: Solve for y first
• Substitution Method: Solve for x first
7.2
Solving Linear Systems by Substitution
1. Solve one of the equations for one of its variables.
2. Substitute the expression from step 1 into the other equation &
solve for the other variable.
3. Substitute the value from step 2 into the revised equation from
step 1 & solve.
4. Check the solution in each of the original equations.
7.3
Solving Linear Systems by Linear Combinations
 Sometimes it is not easy to isolate one of the variables in a linear
system. In that case it may be easier to solve the system by linear
combinations. A linear combination of two equations is an
equation obtained by (1) multiplying one or both equations by a
constant if necessary & (2) adding the resulting equations.
 Sometimes you can solve by adding the original equations because
the coefficients of a variable are already opposites.
7.3
Solving Linear Systems by Linear Combinations
1. Arrange the equations with like terms in columns
2. Multiply, if necessary, the equations by numbers to obtain
coefficients that are opposites for one of the variables.
3. Add the equations from step 2. Combining like terms with
opposite coefficients will eliminate one variable. Solve for the
remaining variable.
4. Substitute the value obtained in step 3 into either of the original
equations & solve for the other variable.
5. Check the solution in each of the original equations.
7.4
Ways to Solve a System of Linear Equations
 Substitution requires that one of the variables be isolated on one
side of the equation. It is especially convenient when one of the
variables has a coefficient or 1 or -1.
 Linear combinations can be applied to any system, but it is
especially convenient when a variable appears in different
equations with coefficients that are opposites.
 Graphing can provide a useful method for estimated a solution.
7.5
Special Types of Linear Systems
 Parallel Lines – If the two equations have the same slope but
different y-intercepts, then the system has no solution
 Lines coincide – If the two equations have the same slop & the
same y-intercept, then the system has infinitely many solutions
 Lines intersect – If the two equations have different slopes, then
the system has one solution
• If the slope is inverse & opposite then the lines are
perpendicular.
7.6
System of Linear Inequalities
 Two or more linear inequalities in the same variables form a
system of linear inequalities, or a system of inequalities. A
solution of a system of linear inequalities in two variables is an
ordered pair that is a solution of each inequality in the system.
1. Graph the boundary lines of each inequality. Use a dashed line
if the inequality is > or < & a solid line if the inequality is ≤ & ≥
2. Shade the appropriate half-plane for each inequality
3. Identify the solution of the system of inequalities as the
intersection of the half-planes from step 2