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System of Linear Equations
Section 4.1
Consider this problem
A roofing contractor bought 30 bundles of shingles
and four rolls of roofing paper for $528. A second
purchase (at the same prices) cost $140 for eight
bundles of shingles and one roll of roofing paper.
Find the price per bundle of shingles and the price
per roll of paper.
This is a system of equations
Analyze the Problem
System of equations: Two or more equations with
two or more variables
How would we solve this problem?
verbal model
Cost of 30 bundles + Cost of 4 rolls = 528
(Frist purchase)
Cost of 8 bundles + Cost of 1 roll = 140
(Second purchase)
System of Linear Equations
If we let 'x' be the prince (in dollars) per bundle of
shingles and y be the price (in dollars) per roll of
paper, we obtain the folloewing system of
equations
30x + 4y = 528 (equation 1)
8x + y = 140
(equation 2)
Solution is a point (x,y)
System of Linear Equations
Solution is a point (x,y)
A solution of such a system is an ordered pair
(x,y) of real numbers that satisfies each equation
in the system. When we find the set of all
solutions of the sutem of equation, we say that we
are solving the system of equations
Real World Application
Ways to Solve Sys. of Eq.
Plug In
Plug In
Decide whether the given ordered pair is a
solution ofthe given system
5x – 4y = 34
x – 2y = 8
a (0,3)
b(6,-1)
Plug In
You determine if a given point is a solution
by plugging it into both equations.
If the answer is true for both equations then
the point is a solution
Steps
1) Plug into both equations
2) Solution only if true for both eq.
Plug In
eq 1
5x – 4y = 34
plug into first equation
eq 2
x – 2y = 8
5(0) – 4(3) = 34
a. (0,3)
0 – 12 = 34
False
is 'a' a solution?
(0,3) Not a solution
(you can stop here)
-12 ≠34
Plug In
eq1 5x – 4y = 34
eq 2
x – 2y = 8
is b a solution
plug into first equation
b(6,-1)
5(6) – 4(-1) = 34
is b a solution
30 + 4 = 34
34 = 34
True for eq 1
Plug In
eq1 5x – 4y = 34
eq 2
x – 2y = 8
is b a solution
b(6,-1)
plug into second
equation
is b a solution
6 – 2(-1) = 8
True for eq1
6+2= 8
Now check eq 2
8=8
True for eq 2
(6, -1) is a solution
Plug In
eq1 x + 2y = 9
plug into first equation
eq2 -2x+3y = 10
x + 2y = 9
a. (1,4)
1 + 2(4) = 9
1+8=9
Determine if 'a' is a
solution
9=9 True for eq 1
Substitution
eq1 x + 2y = 9
eq2 -2x+3y = 10
plug into second
equation
a. (1,4)
-2x+3y = 10
-2(1) + 3(4) = 10
Determine if 'a' is a
solution
-2 + 12 = 10
10 = 10
True for eq 2
a (1, 4) is a solution
Plug In
eq1 x + 2y = 9
plug into first equation
eq2 -2x+3y = 10
x + 2y = 9
b. (-3, 1)
-3 + 2(1) = 9
Determine if 'b' is a
solution
-3 + 2 = 9
-1 = 9
False
Not a solution ( you
may stop here)
Plug In
Try with a partner
One person work 'a' the
other work 'b'
-5x – 2y = 23
x + 4y = -19
a. (-3, -4)
b. (3, 7)
Plug In
Try with a partner
Determine if 'a' is a
solution
One person work 'a' the
other work 'b'
plug into first equation
-5x – 2y = 23
-5(-3) – 2(-4) = 23
x + 4y = -19
15 + 8 = 23
a. (-3, -4)
23 = 23
b. (3, 7)
True for eq 1
Plug In
Try with a partner
plug into second
equation
One person work 'a' the
other work 'b'
-3 + 4(-4) = -19
eq1 -5x – 2y = 23
-3 + -16 = -19
eq2 x + 4y = -19
-19 = -19
a. (-3, -4)
True for eq 2
b. (3, 7)
a is a solution to the
system of equations
Plug In
Try with a partner
plug into first equation
One person work 'a' the -5(3) – 2(7) = 23
other work 'b'
-15 – 14 = 23
eq1 -5x – 2y = 23
-19 = 23
eq2 x + 4y = -19
a. (-3, -4)
b. (3, 7)
Determine if 'b' is a
solution
False (if one is false
you can stop)
b is not a solution to the
system of linear
equations
System of Equations
Substitution Method
Substitution Method
Method of Substitution
1. Solve one equation for one variable in terms of the
other variable
2.Substitute the expression found in step 1 into the other
equation to obtain an equation of one variable
3. Solve the equation obtained in Step 2.
4. Back-substitute the solution from Step 3 into the
expression obtained in Step 1 to find the value of the
other variable
5. Check the solution to see that it satisfies each of the
original equations
Substitution Method
Solve the given system by the substitution
method
eq1
x+y=3
eq2
2x – y = 0
eq 1
x+y=3
(chose a variable to isolate: y)
y = 3 – x ( y is isolated)
Substitution Method
Now plug this transformed eq1 into eq2
eq2
2x – y = 0
2x – (3 – x) = 0
(plug in for y)
2x -3 + x = 0
(dist the negative)
3x – 3 = 0
(combine like terms)
3x = 3
(isolate x-variable)
x=1
(divide both sides by 3)
Substitution Method
x=1
Now back substitute in the equation of your
choice
eq2
2x – y = 0
2(1) – y = 0
2–y=0
-y = -2 (subtract 2 from both sides)
y = 2 (multiply both sides by a negative 1)
Substitution Method
eq 2 (1,2)
Check to see if this works for eq 1
eq1
x+y=3
1+2=3
3=3
(1,2) is the solution to the system of linear
equations
substitution method
Solve the given system by the substitution
method
eq1 x + y = 2
eq2 x – 4y = 12
eq1 x + y = 2
(chose a variable to isolate: x)
x=2–y
Substitution Method
x=2–y
plug in modified eq1 into eq2
x – 4y = 12
(2 – y) – 4y = 12
solve for y
2 – y – 4y = 12
2 – 5y = 12
-5y = 12 – 2
- 5y = 10
Substitution Method
y = -2
Now Back substitute in the equation of your
choice
eq 2
x – 4(-2) = 12
x + 8 = 12
x=4
(4, -2)
(4, -2)
See if this is true for eq 1
eq1 4 + (-2) = 2
4–2=2
2=2
(4, -2) is a solution to the system of linear
equations
Substitution Method
Inconsistent
Inconsistent
Solve the given system by the substitution method
Inconsistent means (no solution)
A problem is inconsistent when the substitution
results in a false statement
ie 2 = 0
(False)
Inconsistent
eq1 y = -4x
eq2 8x + 2 y = 4
plug in modified eq1
into eq2
eq1 y = -4x
8x + 2(-4x) = 4
(chose a variable to
isolate: y)
8x – 8x = 4
0 = 4 (False)
y = -4x
Inconsistent
Substitution Method
Dependent (infinte many solutions)
Dependent
Solve the given system by the substitution method
Dependent means infinite many solutions
A problem is inconsistent when the substitution
results in a false statement
ie 0 = 0
(True)
Dependent
eq1 y = 3x + 4
plug into eq2
eq2 -2y = -6x – 8
-2 (3x + 4) = -6x – 8
eq1 already solved for y -6x – 8 = -6x – 8 (add
6x to both sides)
-8 = -8 True
Dependent (infinete
many solutions)
System of Equations
●
Write the equation in slope intercept form and
then tell how many solutions the system has.
Do not solve
●
eq1 -x + 2y = 8
●
eq2 4x – 8y = 1
●
System of Equations
●
eq2 4x – 8y = 1
●
put in slope-intercept form
●
-8y = -4x + 1
●
eq1
●
y = (½)x – (1/8)
●
-x + 2y = 8
●
2y = x + 8
●
equations are parallel why?
●
Slope is the same
●
●
●
y = (1/2)x + 4
Try this one
●
eq1 5x = -2y + 1
●
eq2 10x = -4y + 2
●
eq2 10x = -4y + 2
●
-4y + 2 = 10x
●
-4y = 10x – 2
●
y = (-5/2)x + (½)
●
put in slope-intercept
form
●
●
eq1 5x = -2y + 1
●
-2y + 1 = 5x
●
-2y = 5x -1
●
y = (-5/2)x + ( ½)
●
equal lines
●
dependent