Transcript 5.1 Solving Systems of Linear Equations by Graphing

5.1 Solving Systems
of Linear Equations
by Graphing
System of Linear Equations


System of Linear Equations

A set of two or more linear equations in the same variables

Example:
x+y=7
Equation 1
2x – 3y = -11
Equation 2
Solution of a system of Linear Equations

In two variables, is an ordered pair that is the solution to
each equation in the system
Example 1: Checking Solutions
Tell whether the ordered pair is a solution of the system of
linear equations.
a)
(2,5); x + y = 7
2x – 3y = -11
Equation 1
Equation 2
Substitute 2 in for x and 5 in for y in each equation.
Equation 1
x+y=7
2+5=7
7=7
Equation 2
2x – 3y = -11
2(2) – 3(5) = -11
4 – 15 = -11
-11 = -11
Since the ordered pair (2,5) is a solution of each of the
equations, it is a solution to the linear system.
Example 1: Checking Solutions
Tell whether the ordered pair is a solution of the system of
linear equations.
b) (-2,0); y = -2x – 4
y=x+4
Equation 1
Equation 2
Substitute -2 in for x and 0 in for y in each equation.
Equation 1
y = -2x - 4
0 = -2(-2) - 4
0=4-4
0=0
Equation 2
y=x+4
0 = -2 + 4
0=2
Since the ordered pair (-2,0) is not a solution of each of
the equations, it is NOT a solution to the linear system.
You try!
Tell whether the ordered pair is a solution to the system of
linear equations.
1)
(1, -2); 2x + y = 0
-x + 2y = 5
2)
(1,4); y = 3x + 1
y = -x + 5
Equation 1
2(1) + (-2) = 0
2 + (-2) = 0
0=0
Equation 1
4 = 3(1) + 1
4=3+1
4=4
Equation 2
-(1) + 2(-2) = 5
-1 + (-4) = 5
-5 = 5
Equation 2
4 = -(1) + 5
4 = -1 + 5
4=4
(1,-2) is not a solution to the
system of linear equations.
(1,4) is a solution to the
system of linear equations.
Solving Systems of Linear
Equations by Graphing
Step 1: Graph each equation in the same coordinate plane.
Step 2: Estimate the point of intersection.
Step 3: Check the point from step 2 by substituting for x
and y in each equation of the original system.
Example 2: Solving a System of
Linear Equations by Graphing
a) Solve the system of Linear equations by graphing:
y = -2x + 5
Equation 1
y = 4x – 1
Equation 2
2) The two lines
intersect at point (1, 3).
1)
3) Check your work:
y = -2x + 5
3 = -2(1) + 5
3=3
Equation 1
y = 4x – 1
Equation 2
3 = 4(1) – 1
3=3
(1,3) is the solution to the system.
Example 2: Solving a System of
Linear Equations by Graphing
b) Solve the system of Linear equations by graphing:
2x + y = 5
Equation 1
3x – 2y = 4
Equation 2
2x + 0 = 5
3x – 2(0) = 4
2x = 5
3x = 4
x = 5/2 = 2 ½
x = 4/3 = 1 1/3
2(0) + y = 5
y=5
1)
3(0) – 2y = 4
-2y = 4
y = -2
2) The two lines intersect at
point (2, 1).
3) Check your work:
2x + y = 5
2(2) + 1 = 5
5=5
Equation 1
3x – 2y = 4
Equation 2
3(2) – 2(1) = 4
6–2=4
4=4
(2,1) is the solution to the system.
You try!
Solve the system of linear equations by graphing.
y=x–2
Equation 1
y = -x + 4
Equation 2
The lines intersect at (3,1).
Remember to
substitute those
values back into your
original equations.
Did they work?
(3,1) is the solution.
Example 3: Solving Real-Life
Problems
A roofing contractor buys 30 bundles of shingles and 4 rolls
of roofing paper for $1040. In a second purchase (at the
same prices), the contractor buys 8 bundles of shingles for
$256. Find the price per bundle of shingles and the price
per roll of roofing paper.
30 ∙ 𝑝𝑟𝑖𝑐𝑒 𝑝𝑒𝑟 𝑏𝑢𝑛𝑑𝑙𝑒 + 4 ∙ 𝑝𝑟𝑖𝑐𝑒 𝑝𝑒𝑟 𝑟𝑜𝑙𝑙 = 1040
8 ∙ 𝑝𝑟𝑖𝑐𝑒 𝑝𝑒𝑟 𝑏𝑢𝑛𝑑𝑙𝑒 + 0 ∙ 𝑝𝑟𝑖𝑐𝑒 𝑝𝑒𝑟 𝑟𝑜𝑙𝑙 = 256
Let x be the price (in dollars) per bundle.
Let y be the price (in dollars) per roll.
𝑆𝑦𝑠𝑡𝑒𝑚 ∶ 30𝑥 + 4𝑦 = 1040
8𝑥 = 256
𝑆𝑦𝑠𝑡𝑒𝑚 ∶ 30𝑥 + 4𝑦 = 1040
8𝑥 = 256
𝑆𝑦𝑠𝑡𝑒𝑚 ∶ 𝑦 = −7.5𝑥 + 260
𝑥 = 32
It looks as if the lines
intersect at (32,20). This
would mean that the price
per bundle of shingles is $32
and the price per roll of
roofing paper is $20.
Equation 1
Equation 2
Equation 1
Equation 2
Remember you can check your solution by substituting the
x and y values into the original equations!!
Equation 1
30x + 4y = 1040
30(32) + 4(20) = 1040
1040 = 1040
Equation 2
8x = 256
8(32) = 256
256 = 256
You try!
You have a total of 18 math and science exercises for
homework. You have 6 more math exercises than science
exercises. How many exercises do you have in each
subject?
(# of math exercises) + (# of science exercises) = 18
(# of math exercises) – 6 = # of sciences exercises
Let x be the number of math exercises.
Let y be the number of science exercises.
𝑆𝑦𝑠𝑡𝑒𝑚 ∶ 𝑥 + 𝑦 = 18
𝑥−6=𝑦
𝑆𝑦𝑠𝑡𝑒𝑚 ∶ 𝑥 + 𝑦 = 18
𝑥−6=𝑦
The lines intersect
at (12, 6). This
means that you
have 12 math
exercises and 6
science exercises.
Check your solution!