78910SytemsofEquations

Question 1

What is the solution to a system of equations?

Answer 1

An ordered pair that solves all equations.

This is where the lines intersect.

If there is no solution, they don’t intersect.

Question 2

How do you solve a simple 2-equation system?

y = 2x - 5

y = x - 2

Answer 2

Set them equal to each other.

y = 2x - 5; y = x - 2

2x - 5 = x - 2

x = 3

:: y = 1 in both equations

(3, 1)

Question 3

How many solutions can systems of linear equations have?

Answer 3

0 (if they don’t intersect) or 1.

They are straight lines and can only intersect once.

Question 4

Solve:

3x + 2y = 5

6x + 4y = 19

Answer 4

3x + 2y = 5 →

y - ^-3/₂x + 5

6x + 4y = 19 →

y = ^-3/₂x + 19

No solution. They are parallel (same slope).

Question 5

What else is the solution to a system of equations called?

Answer 5

An Independent Solution.

Question 6

What is another way to solve systems of equations?

Answer 6

Substitution.

Solve for one of the variables in one of the equations and plug

Question 7

Solve, using substitution:

2x + 3y = 4

-6x + y = -7

Answer 7

2x + 3y = 4

-6x + y = -7 → y = 6x - 7

2x + 3(6x - 7) = 4

20x = 25

x = ⁵/₄

y = 6(5/4) - 7 = ½

(Check both equations)

Question 8

What is another way to solve a system of equations?

Answer 8

Elimination.

You can eliminate a variable by addition, subtraction, multiplication or division.

Question 9

Solve by elimination:

2x - 5y = 12

6x + 5y = 36

Answer 9

Use addition:

2x - 5y = 12

6x + 5y = 36

8x = 48

x = 6

:: y = 0 (check both equations)

Question 10

Solve by elimination:

5s + 4t = 27

3s + 4t = 31

Answer 10

Use subtraction:

5s + 4t = 27

3s + 4t = 31

2s = -4 → s = -2

:: -10 + 4t = 27 → 4t = 37

t = 37/4

Question 11

How do you solve a system of 3 linear equations with 3 variables?

Answer 11

Play with using 2 equations to eliminate a variable.

Question 12

Solve:

^#1: 2x + 3y - z = 15

^#2: x - 3y + 3z = -4

^#3: 4x - 3y - z = 19

Answer 12

^#1: 2x + 3y - z = 15

^#2: x - 3y + 3z = -4

^#3: 4x - 3y - z = 19

^{add #1 and #2 to eliminate y:}

^#1+#2: 3x + 2z = 11

^{add #1 and #3 to eliminate y:}

^#1+#3: 6x - 2z = 34

^{Add these 2 to eliminate z:}

9x = 45 → x = 5

^{Plug into one of the 2 equations:}

3(5) + 2z = 11 → z = 2

:: y = 1

Question 13

Use elimination by multiplication to solve:

^#1: 8x + 2y - 4z = 0

^#2: 2x - 3y + 3z = 9

^#3: -6x -2y + z = 0

Answer 13

^{#1: 8x + 2y - 4z = 0}

^{#2: 2x - 3y + 3z = 9}

^{#3: -6x -2y + z = 0}

^{Multiply #3 by 4 → -24x - 8y + 4z = 0 eliminate z}

^{add to #1 → -16x -6y = 0}

^{Multiply #3 by -3 → 18x + 6y - 3z = 0}

^{add to #2 → 20x + 3y = 9 eliminate z}

^{You now have 2 equations with 2 variables. Set them equal:}

^{-16x -6y = 20x + 3y - 9 Solve for y: y = -4x + 1}

^{Plug into 2-variable equation: x = ¾ y = -2}

^{Plug into original equation: z = ½}

Question 14

How do you graph linear equalities?

Answer 14

Graph both lines and mark the side for the inequalities and see where they intersect.