15.1 solving systems of equations by graphing

Slide - 1Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
2
Systems of Linear
Equations and
Inequalities
15

1. Decide whether a given ordered pair is a
solution of a system.
2. Solve linear systems by graphing.
3. Solve special systems by graphing.
4. Identify special systems without graphing.
Objectives
15.1 Solving Systems of Linear Equations by
Graphing

A system of linear equations, often called a linear system,
consists of two or more linear equations with the same
variables.
Decide Whether a Given Ordered Pair is a Solution
2x + 3y = 4
3x – y = –5
or x + 3y = 1
–y = 4 – 2x
or x – y = 1
y = 3
A solution of a system of linear equations is an ordered pair
that makes both equations true at the same time. A solution of
an equation is said to satisfy the equation.

Example Is (2,–1) a solution of the system 3x + y = 5
2x – 3y = 7 ?
Substitute 2 for x and –1 for y in each equation.
3(2) + (–1) = 5 ?
6 – 1 = 5 ?
5 = 5 True
2(2) – 3(–1) = 7 ?
4 + 3 = 7 ?
7 = 7 True
Since (2,–1) satisfies both equations, it is a solution of the
system.

Example Is (2,–1) a solution of the system x + 5y = –3
4x + 2y = 1 ?
Substitute 2 for x and –1 for y in each equation.
2 + 5(–1) = – 3?
2 – 5 = –3?
–3 = –3 True
4(2) + 2(–1) = 1?
8 – 2 = 1?
6 = 1 False
(2,–1) is not a solution of this system because it does not
satisfy the second equation.

Example
Solve the system of equations by graphing both equations on
the same axes.
Rewrite each equation in slope-intercept form to graph.
Solve Linear Systems by Graphing
–2x + ⅔y = –4 becomes y = 3x – 6 y-intercept (0, – 6); m = 3
5x – y = 8 becomes y = 5x – 8 y-intercept (0, – 8); m = 5
2
2 4
3
5 8
x y
x y
   
 

Example (cont)
Graph both lines on the same axes and identify where
they cross.
y = 3x – 6 y = 5x – 8
-9
-8
-7
-6
-5
-3
-1
-4
-2
1
42-2-4 531-1-3-5
Because (1,–3) satis-
fies both equations,
the solution set of this
system is {(1,–3)}.

CAUTION
We recommend using graph paper and a straightedge when
solving systems of equations graphically. It may not be
possible to determine from the graph the exact coordinates of
the point that represents the solution, particularly if those
coordinates are not integers. The graphing method does,
however, show geometrically how solutions are found and is
useful when approximate answers will suffice.

Example
Solve each system by graphing.
(a) 3x + y = 4
6x + 2y = 1
3x + y = 4 becomes y = –3x + 4;
y-intercept (0, 4); m = –3
Solve Special Systems by Graphing
6x + 2y = 1 becomes y = –3x + ½
y-intercept (0, ½); m = –3

Example (cont)
The graphs of these
lines are parallel and
have no points in
common. For such a
system, there is no
solution.
y = –3x + 4
y = –3x + ½
-5
-3
-1
-4
-2
1
3
5
2
4
42-2-4 531-1-3-5

Example (cont)
Solve each system by graphing.
(b) ½x + y = 3
2x + 4y = 12
½x + y = 3 becomes y = –½x + 3;
y-intercept (0, 3); m = –½
2x + 4y = 12 becomes y = –½x + 3;
y-intercept (0, 3); m = –½

Example (cont)
The graphs of these two
equations are the same
line. Thus, every point on
the line is a solution of the
system, and the solution
set contains an infinite
number of ordered pairs
that satisfy the equations.
y = – ½x + 3
y = – ½x + 3
-5
-3
-1
-4
-2
1
3
5
2
4
42-2-4 531-1-3-5

Three Cases for Solutions of Linear Systems with
Two Variables
1. The graphs intersect at exactly one point, which
gives the (single) ordered-pair solution of the
system. The system is consistent, and the
equations are independent.

Two Variables (cont)
2. The graphs are parallel lines. So, there is no
solution and the solution set is . The system is
inconsistent and the equations are independent.
0

Two Variables (cont)
3. The graphs are the same line. There is an infinite
number of solutions, and the solution set is written
in set-builder notation. The system is consistent
and the equations are dependent.

Example
Describe the system without graphing. State the number of
solutions
(a) 3x + 2y = 6
–2y = 3x – 5
Identify Special Systems without Graphing
Rewrite each equation in slope-intercept form.
3 2 6
2 3 6
3
3
2
 
  
  
x y
y x
y x
2 3 5
3 5
2 2
  
  
y x
y x

Example (cont)
Both lines have slope but have different y-intercepts, (0,3)
and . Lines with the same slope are parallel, so these
equations have graphs that are parallel lines. Thus, the system
has no solution.
Identify Special Systems without Graphing
3
2

5
0,
2
 
 
 

15.1 solving systems of equations by graphing

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