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 EquationsIf 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

eq 2 x – 2y = 8

a. (0,3)

is 'a' a solution?

(0,3) Not a solution (you can stop here)

plug into first equation

5(0) – 4(3) = 34

0 – 12 = 34

False

-12 ≠34

Plug In

eq1 5x – 4y = 34

eq 2 x – 2y = 8

b(6,-1)

is b a solution

is b a solution

plug into first equation

5(6) – 4(-1) = 34

30 + 4 = 34

34 = 34

True for eq 1

Plug In

eq1 5x – 4y = 34

eq 2 x – 2y = 8

b(6,-1)

is b a solution

True for eq1

Now check eq 2

is b a solution

plug into second equation

6 – 2(-1) = 8

6 + 2 = 8

8 = 8

True for eq 2

(6, -1) is a solution

Plug In

eq1 x + 2y = 9

eq2 -2x+3y = 10

a. (1,4)

Determine if 'a' is a solution

plug into first equation

x + 2y = 9

1 + 2(4) = 9

1 + 8 = 9

9=9 True for eq 1

Substitution

eq1 x + 2y = 9

eq2 -2x+3y = 10

a. (1,4)

Determine if 'a' is a solution

plug into second equation

-2x+3y = 10

-2(1) + 3(4) = 10

-2 + 12 = 10

10 = 10

True for eq 2

a (1, 4) is a solution

Plug In

eq1 x + 2y = 9

eq2 -2x+3y = 10

b. (-3, 1)

Determine if 'b' is a solution

plug into first equation

x + 2y = 9

-3 + 2(1) = 9

-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

One person work 'a' the other work 'b'

-5x – 2y = 23

x + 4y = -19

a. (-3, -4)

b. (3, 7)

Determine if 'a' is a solution

plug into first equation

-5(-3) – 2(-4) = 23

15 + 8 = 23

23 = 23

True for eq 1

Plug In

Try with a partner

One person work 'a' the other work 'b'

eq1 -5x – 2y = 23

eq2 x + 4y = -19

a. (-3, -4)

b. (3, 7)

plug into second equation

-3 + 4(-4) = -19

-3 + -16 = -19

-19 = -19

True for eq 2

a is a solution to the system of equations

Plug In

Try with a partner

One person work 'a' the other work 'b'

eq1 -5x – 2y = 23

eq2 x + 4y = -19

a. (-3, -4)

b. (3, 7)

Determine if 'b' is a solution

plug into first equation

-5(3) – 2(7) = 23

-15 – 14 = 23

-19 = 23

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)

eq 2 (1,2)

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

y = -2

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

eq1 y = -4x

(chose a variable to isolate: y)

y = -4x

plug in modified eq1 into eq2

8x + 2(-4x) = 4

8x – 8x = 4

0 = 4 (False)

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

eq2 -2y = -6x – 8

eq1 already solved for y

plug into eq2

-2 (3x + 4) = -6x – 8

-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

● put in slope-intercept form

● eq1

● -x + 2y = 8

● 2y = x + 8

● y = (1/2)x + 4

●  

●  

● eq2 4x – 8y = 1

● -8y = -4x + 1

● y = (½)x – (1/8)

● equations are parallel why?

● Slope is the same

Try this one

● eq1 5x = -2y + 1

● eq2 10x = -4y + 2

● put in slope-intercept form

● eq1 5x = -2y + 1

● -2y + 1 = 5x

● -2y = 5x -1

● y = (-5/2)x + ( ½)

● eq2 10x = -4y + 2

● -4y + 2 = 10x

● -4y = 10x – 2

● y = (-5/2)x + (½)

●  

● equal lines

● dependent