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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
CONSISTENT AND
INDEPENDENT
CONSISTENT AND
DEPENDENT
INCONSISTENT
Number of Solutions
exactly one
infinitely many
none
Description
different slopes
same slope,
same y-intercept
same slope, different
y-intercept
Graph
𝑦 = 3𝑥 + 4 𝑦 = −3𝑥 + 2
𝑦 = 4𝑥 − 5
𝑦 = 4𝑥 − 5
𝑦 = 7𝑥 + 1
𝑦 = 7𝑥 − 1
𝑦 = −8𝑥 − 3 𝑦 = −8𝑥 − 3
Different slopes CONSISTENT and INDEPENDENT
so there is 1 solution to the system
Different slopes
Same slope,
Same y-intercept
Same slope,
Different y-intercepts
Same Slope, Same
y-intercept
CONSISTENT and INDEPENDENT
so there is 1 solution to the system
CONSISTENT and DEPENDENT
So there are infinite solutions
INCONSISTENT
So there is no solution
CONSISTENT and DEPENDENT
So there are infinite solutions
𝑦 = 5𝑥 + 4 𝑦 = 3𝑥 − 5
𝒚 = 𝒎𝒙 + 𝒃 𝒎 is the slope
𝒃 is the y − intercept
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.
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).
Standard Form:
4𝑥 − 𝑦 = 6 2𝑥 + 𝑦 = 0
Slope-Intercept Form:
𝑦 = 4𝑥 − 6 𝑦 = −2𝑥
(1) y = – 4x
16 = – 4(– 4) 16 = 16
(2) y = – 2x + 8
16 = – 2(– 4) + 8 16 = 8 + 8 16 = 16
(-4,16) is a solution.
Determine if (– 4, 16) is a solution to the system of equations.
y = – 4x y = – 2x + 8
Solution or Not?
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
y = 3x + 3
3 = 3(– 2) + 3 3 = – 6 + 3
3 = – 3
(-2,3) is not a solution.
How to graph a linear equation in two variables?
𝟑𝒙 − 𝒚 = −𝟏
Standard Form 𝒂𝒙 + 𝒃𝒚 = 𝒄
Slope-Intercept Form 𝒚 = 𝒎𝒙 + 𝒃
𝒚 = 𝟑𝒙 + 𝟏
slope (𝒎) =rise
run=
𝟑
𝟏
y − intercept 𝒃 = 𝟏 (0,1)
𝒙 + 𝟐𝒚 = 𝟕
Standard Form 𝒂𝒙 + 𝒃𝒚 = 𝒄
Slope-Intercept Form 𝒚 = 𝒎𝒙 + 𝒃
𝒚 = −𝟏
𝟐𝒙 +
𝟕
𝟐
slope 𝒎 =rise
run= −
𝟏
𝟐
y − intercept 𝒃 =7
2
How to graph a linear equation in two variables?
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.
𝑥 − 𝑦 = 4 𝑥 + 𝑦 = 2 1.
2𝑥 − 𝑦 = −1 𝑥 + 𝑦 = 7
𝑥 − 2𝑦 = −2 3𝑥 − 2𝑦 = 2
2𝑥 + 2𝑦 = 6 4𝑥 − 6𝑦 = 12
2.
3.
4.
5.
6.
2𝑥 + 𝑦 = −1 𝑥 − 𝑦 = −5
3𝑥 − 2𝑦 = 8 𝑥 + 𝑦 = 6
𝒙 − 𝒚 = 𝟒
𝒙 + 𝒚 = 𝟐
Solving Systems of Linear Equations
by Graphing
Point of Intersection: (3,-1)
𝒙 + 𝒚 = 𝟕
𝟐𝒙 − 𝒚 = −𝟏
Solving Systems of Linear Equations
by Graphing
Point of Intersection: (2,5)
𝟑𝒙 − 𝟐𝒚 = 𝟐 𝒙 − 𝟐𝒚 = −𝟐
Solving Systems of Linear Equations
by Graphing
Point of Intersection: (2,2)
𝟒𝒙 − 𝟔𝒚 = 𝟏𝟐
𝟐𝒙 + 𝟐𝒚 = 𝟔
Solving Systems of Linear Equations
by Graphing
Point of Intersection: (3,0)
𝟐𝒙 + 𝒚 = −𝟏 𝒙 – 𝒚 = −𝟓
y = x + 5
y = –2x – 1
Solving Systems of Linear Equations
by Graphing
Point of Intersection: (-2,3)
𝟑𝒙 − 𝟐𝒚 = 𝟖 𝒙 + 𝒚 = 𝟔
Solving Systems of Linear Equations
by Graphing
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.
Step 2: Graph both equations
on the same coordinate plane.
Step 3: Look for the point
of intersection.
Step 4: Check to make sure your
solution makes both equations
true.
Solve both equations for y, so
that each equation looks like
𝑦 = 𝑚𝑥 + 𝑏.
Use the slope and 𝑦-intercept for
each equation in step 1.
This ordered pair that corresponds to the point of intersection is the solution.
Substitute the 𝑥 and 𝑦 values
into both equations to verify
the point is a solution to both
equations.
Solve the system by graphing. Check your answer.
𝒚 = 𝒙 𝒚 = −𝟐𝒙 – 𝟑
1. Rewrite the equations in
slope-intercept form.
(–1,–1) is the solution of the system.
3. Check..
•
𝒚 = 𝒙
(–1) (–1)
–1 –1
𝒚 = – 𝟐𝒙 – 𝟑
(–1) –2(–1) –3
–1 2 – 3
–1 – 1
2. Graph the system.
𝒙 − 𝒚 = 𝟎 𝟐𝒙 + 𝒚 = – 𝟑
Solving Systems of Linear Equations
by Graphing
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.
4𝑥 − 7𝑦 = −35 2𝑥 + 7𝑦 = −7
1.
2𝑥 − 3𝑦 = −3 𝑥 + 𝑦 = −4
3𝑥 − 2𝑦 = 4 3𝑥 − 𝑦 = 5
𝑥 − 𝑦 = 1 𝑥 + 3𝑦 = 9
2.
3.
4.
5.
6.
4𝑥 − 3𝑦 = −15 𝑥 − 3𝑦 = −6
4𝑥 + 𝑦 = 4 3𝑥 − 𝑦 = 3
−5𝑥 + 4𝑦 = −16 𝑥 + 4𝑦 = 8
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 .
4𝑥 + 9𝑦 = −27 7𝑥 + 5𝑦 = −15
ASSESSMENT
1.
2.
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.
ASSIGNMENT
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
