Solving Systems of Linear
Equations in Two Variables
by Graphing
Presented by:
JOEY F. VALDRIZ
LEARNING COMPETENCY:
Solve a system of linear
equations in two variables
by graphing.
Code: M8AL-Ii-j-1
RECALL
What are the types of
systems of linear equations
in two variables?
TYPES OF SYSTEMS OF LINEAR EQUATIONS
Classification
Number of
Solutions
Description
Graph
CONSISTENT AND
INDEPENDENT
CONSISTENT AND
DEPENDENT
INCONSISTENT
exactly one
infinitely many
none
different slopes
same slope,
same y-intercept
same slope,
different
y-intercept
𝒚 = 𝒎𝒙 + 𝒃
𝒎 is the slope
𝒃 is the y − intercept
𝑦 = 3𝑥 + 4 Different slopes
𝑦 = −3𝑥 + 2
CONSISTENT and INDEPENDENT
so there is 1 solution to the system
𝑦 = 5𝑥 + 4
𝑦 = 3𝑥 − 5
Different slopes
CONSISTENT and INDEPENDENT
so there is 1 solution to the system
𝑦 = 4𝑥 − 5
𝑦 = 4𝑥 − 5
Same slope,
Same y-intercept
𝑦 = 7𝑥 + 1
𝑦 = 7𝑥 − 1
𝑦 = −8𝑥 − 3
𝑦 = −8𝑥 − 3
Same slope,
Different y-intercepts
Same Slope, Same
y-intercept
CONSISTENT and DEPENDENT
So there are infinite solutions
INCONSISTENT
So there is no solution
CONSISTENT and DEPENDENT
So there are infinite solutions
How many lines appear below?
Unlocking of Difficulties
A system of linear equations is two or more linear
equations whose solution we are trying to find.
Standard Form:
4𝑥 − 𝑦 = 6
2𝑥 + 𝑦 = 0
Slope-Intercept Form:
𝑦 = 4𝑥 − 6
𝑦 = −2𝑥
A solution to a system of linear equations in two
variables is the ordered pair (𝑥, 𝑦)that satisfies all
equations in the system. The solution to the above
system is (1, – 2).
Solution or Not?
Determine if (– 4, 16) is a solution to the
system of equations.
y = – 4x
y = – 2x + 8
(1)
y = – 4x
16 = – 4(– 4)
16 = 16
(-4,16) is a
solution.
(2)
y = – 2x + 8
16 = – 2(– 4) + 8
16 = 8 + 8
16 = 16
Solution or Not?
 Determine if (– 2, 3) is a solution to the
system of equations.
𝒙 + 𝟐𝒚 = 𝟒
𝒚 = 𝟑𝒙 + 𝟑
x + 2y = 4
– 2 + 2(3) = 4
–2+6=4
4=4
(-2,3) is not
a solution.
y = 3x + 3
3 = 3(– 2) + 3
3=–6+3
3=–3
How to graph a linear equation in two variables?
Standard Form
𝒂𝒙 + 𝒃𝒚 = 𝒄
𝟑𝒙 − 𝒚 = −𝟏
Slope-Intercept Form
𝒚 = 𝒎𝒙 + 𝒃
𝒚 = 𝟑𝒙 + 𝟏
rise 𝟑
slope (𝒎) =
=
run 𝟏
y − intercept 𝒃 = 𝟏
(0,1)
How to graph a linear equation in two variables?
Standard Form
𝒂𝒙 + 𝒃𝒚 = 𝒄
𝒙 + 𝟐𝒚 = 𝟕
Slope-Intercept Form
𝒚 = 𝒎𝒙 + 𝒃
𝟏
𝟕
𝒚=− 𝒙+
𝟐
𝟐
rise
𝟏
slope 𝒎 =
=−
run
𝟐
7
y − intercept 𝒃 =
2
Let’s do this!
Graph the following systems of linear equations in two
variables. Be able to find the point of intersection and
the ordered pair that corresponds to it.
1.
𝑥−𝑦 =4
𝑥+𝑦 =2
2.
2𝑥 − 𝑦 = −1
𝑥+𝑦 =7
3.
𝑥 − 2𝑦 = −2
3𝑥 − 2𝑦 = 2
4.
2𝑥 + 2𝑦 = 6
4𝑥 − 6𝑦 = 12
5.
2𝑥 + 𝑦 = −1
𝑥 − 𝑦 = −5
6.
3𝑥 − 2𝑦 = 8
𝑥+𝑦 =6
Solving Systems of Linear Equations
by Graphing
𝒙−𝒚=𝟒
𝒙+𝒚 = 𝟐
Point of Intersection: (3,-1)

Solving Systems of Linear Equations
by Graphing
𝟐𝒙 − 𝒚 = −𝟏
𝒙+𝒚 = 𝟕
Point of Intersection: (2,5)
Solving Systems of Linear Equations
by Graphing
𝒙 − 𝟐𝒚 = −𝟐
𝟑𝒙 − 𝟐𝒚 = 𝟐
Point of Intersection: (2,2)
Solving Systems of Linear Equations
by Graphing
𝟐𝒙 + 𝟐𝒚 = 𝟔
𝟒𝒙 − 𝟔𝒚 = 𝟏𝟐
Point of Intersection: (3,0)
Solving Systems of Linear Equations
by Graphing
𝟐𝒙 + 𝒚 = −𝟏
𝒙 – 𝒚 = −𝟓
y = x +5
y = –2x – 1
Point of Intersection: (-2,3)
Solving Systems of Linear Equations
by Graphing
𝟑𝒙 − 𝟐𝒚 = 𝟖
𝒙+𝒚=𝟔
Point of Intersection: (4,2)
Solving a System of Linear Equations in Two
Variables by Graphing
There are four steps to solving a linear system using a graph:
Step 1: Put both equations in
slope-intercept form.
Solve both equations for y, so
that each equation looks like
𝑦 = 𝑚𝑥 + 𝑏.
Step 2: Graph both equations
on the same coordinate plane.
Use the slope and 𝑦-intercept for
each equation in step 1.
Step 3: Look for the point
of intersection.
This ordered pair that
corresponds to the point of
intersection is the solution.
Step 4: Check to make sure your
solution makes both equations
true.
Substitute the 𝑥 and 𝑦 values
into both equations to verify
the point is a solution to both
equations.
Solving Systems of Linear Equations
by Graphing
Solve the system by graphing. Check your answer.
𝒙−𝒚=𝟎
𝟐𝒙 + 𝒚 = – 𝟑
1. Rewrite the equations in
slope-intercept form.
2. Graph the system.
3. Check..
𝒚 = 𝒙
𝒚 = −𝟐𝒙 – 𝟑
𝒚 = 𝒙
•
(–1)
–1
(–1)
–1
(–1,–1) is the solution of the system.
𝒚 = – 𝟐𝒙 – 𝟑
(–1)
–1
–1
–2(–1) –3
2–3
–1
Application
Solve each of the following systems of linear equations
in two variables. Then, identify the name of the
barangay on the map where the solution is found. You
have to tell something about the barangay afterward.
1.
4𝑥 − 7𝑦 = −35 5. 4𝑥 − 3𝑦 = −15
𝑥 − 3𝑦 = −6
2𝑥 + 7𝑦 = −7
2. 2𝑥 − 3𝑦 = −3
𝑥 + 𝑦 = −4
3.
3𝑥 − 2𝑦 = 4
3𝑥 − 𝑦 = 5
4.
𝑥−𝑦 =1
𝑥 + 3𝑦 = 9
6.
4𝑥 + 𝑦 = 4
3𝑥 − 𝑦 = 3
ASSESSMENT
Graph the following systems of linear equations
in two variables using one coordinate plane.
Label the solution. In transforming the linear
equations from standard form to slope-intercept
form, you may use the back portion of your
graphing paper .
1.
−5𝑥 + 4𝑦 = −16
𝑥 + 4𝑦 = 8
2.
4𝑥 + 9𝑦 = −27
7𝑥 + 5𝑦 = −15
ASSIGNMENT
Analyze the following graphs of systems of linear equations in
two variables. Write a system of linear equations in two
variables represented by each of the graphs. Use standard
form (𝑎𝑥 + 𝑏𝑦 = 𝑐) in writing your linear equations.
1.
2.
3.
“Life is not linear; you
have ups and downs.
It’s how you deal with
the troughs that
defines you.”
~Michael Lee-Chin