Algebra 1 Common Core Student Edition, Grade 8-9

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If tw o equations have
the same slope and
y-intercept, their graphs
w ill be th e same line. If
tw o equations have the
same slope but different
y-intercepts, their graphs
w ill be p a ra lle l lines.

Pr o b l em 3 Syst ems With Infinitely M any Solut ions or No Solution

What is the solution of each system? Use a graph.

Q 2y — x = 2

y = ox^1 + 1_L_ 1

G raph the equations 2y - x = 2 and y = | x + 1 in th e sam e

coordinate plane.

The equations rep resen t the sam e line. Any p o in t on th e line is a

solution of th e system, so there are infinitely m any solutions. The

system is consistent an d d ep en d en t.

Qy = 2x + 2

y = 2x - 1

Graph the equations y = 2x + 2 an d y = 2x - 1 in th e sam e

coordinate plane.

The lines are parallel, so there is no solution. The system is inconsistent.

sj Go t I t? 3. W hat is the solution of each system in p arts (a) an d (b)? Use

a graph. D escribe th e n u m b e r of solutions,

a. y = —x — 3 b. y = 3 x - 3

y = —x + 5 3y = 9x — 9

c. R e a s o n in g Before graphing th e equations, how can you d eterm in e

w h e th e r a system of equations h as exactly one solution, infinitely m any

solutions, or no solution?

Co n cep t Su m m a r y Syst ems of Li near Equat i ons

One solution Infinitely many solutions No solution

The lines intersect at one

point. The lines have

different slopes. The

equations are consistent

an d in d ep en d en t.

The lines are th e same.

The lines have th e sam e

slope an d y-intercept. The

equations are consistent

and dependent.

The lines are parallel. The

lines have the sam e slope

an d different y-intercepts.

The eq u atio n s are

inconsistent.

366 Chapter 6 Systems of Equations and Inequalities