SECTION 4.1
Systems of Linear Equations in Two Variables
EXAMPLE 1
Determine if the ordered pair is a solution to the
system.
2𝑥 − 𝑦 = 8
5, 2 ;
3𝑥 + 2𝑦 = 20
SYSTEMS OF LINEAR EQUATIONS
Possible Solutions to Linear Systems in Two Variables
Number of
Solutions
Terminology
Graphically
One
Consistent,
Independent
Lines intersect at
one point.
Infinitely Many
Consistent,
Dependent
The lines are the
same.
No Solution
Inconsistent,
Independent
The lines are
parallel.
EXAMPLE 2
Solve the system by substitution. If it’s
inconsistent or dependent, say so.
2𝑥 − 𝑦 = 6
𝑦 = 5𝑥
SUBSTITUTION METHOD
Substitution Method Summary
Step 1
Step 2
Isolate a variable.
Plug what you get into the other
equation.
Step 3
Solve the new equation you get in
Step 2.
Plug the solution from the Step 2
equation back into the first equation.
Interpret your solution.
Step 4
Step 5
EXAMPLE 3
Solve the system by substitution. If it’s
inconsistent or dependent, say so.
−3𝑥 + 𝑦 = −5
𝑥 + 2𝑦 = 0
A 2.3 PROBLEM REVISITED . . .
Example 1 (#32)
The perimeter of a certain rectangle is 16
times the width. The length is 12 cm
more than the width. Find the length and
width of the rectangle.
EXAMPLE 4
Solve the system by substitution. If it’s
inconsistent or dependent, say so.
𝑦 = −4𝑥
8𝑥 + 2𝑦 = 4
EXAMPLE 5
Solve the system by elimination. If it’s inconsistent
or dependent, say so.
6𝑥 + 5𝑦 = −7
−6𝑥 − 11𝑦 = 1
ELIMINATION METHOD
Elimination Method Summary
Step 1
Step 2
Step 3
Step 4
Step 5
Step 6
Write both equations in 𝐴𝑥 + 𝐵𝑦 = 𝐶
form.
Focus on eliminating one of the
variables by multiplying by a
constant.
Add the new equations.
Solve the equation obtained in step 3.
Plug what you get into an original
equation.
Interpret your answer.
EXAMPLE 6
Solve the system by elimination. If it’s inconsistent
or dependent, say so.
2𝑥 − 3𝑦 = 7
−4𝑥 + 6𝑦 = 14
EXAMPLE 7
Solve the system by elimination. If it’s inconsistent
or dependent, say so.
4𝑥 + 3𝑦 = 1
3𝑥 + 2𝑦 = 2
EXAMPLE 8
Solve the system by elimination. If it’s inconsistent
or dependent, say so.
3
𝑥 + 𝑦=3
2
2
1
𝑥+ 𝑦=1
3
3
AN APPLICATION . . .
Solve the problem.
Two positive numbers add to be 131. Their difference
is 55. Find the numbers.
EXAMPLE 9
Write in slope-intercept form and tell how many
solutions it has.
−𝑥 + 2𝑦 = 8
4𝑥 − 8𝑦 = 1
QUESTIONS???
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