Linear Systems

Question 1

What is the equation for a vertical line?

Answer 1

x=x Intercept

Example: x=3

Question 2

What is the equation for a horizontal line?

Answer 2

y=y Intercept

Example: y=2

Question 3

What is an Intercept?

Answer 3

Where a line segment crosses the x or y axis.

Question 4

How do you use a table of values to determine all of the points in a line segment?

Answer 4

Step 1) Choose your X values

Step 2) Write out your equations to find Y and replace the x variables with the values you chose

Step 3) Write your x and y coordinate pair

Question 5

How do you plot an equation in y=mx+b form?

Answer 5

Step 1) Plot the y-intercept (b)

Step 2) From the y-intercept, use rise/run to find another point

Example: y=2x+4

Plot (0,4)

Use rise/run to plot the next point

m= 2/1 (up 2 over 1)

Question 6

How do you find the x and y intercepts of a line?

Answer 6

Example: y= x+6

Set the opposite intercept to zero and solve. Graph all found points.

X Intercept Y Intercept

y=0 x=0

y=x+6 y=x+6

0=x+6 y=0+6

-6=x y=6

(-6,0) (0,6)

Question 7

What does slope measure?

Answer 7

Steepness, the greater the slope the steeper the line.

Example: y=-10x+3 is steeper than y=5x-7

Question 8

What equation is used to find the slope of a line when given 2 points?

Answer 8

y2-y1

x2-x1

Question 9

What equation is used to find the equation of a line when given slope and the y-intercept?

Answer 9

Substitute the slope and the y-intercept into y=mx+b

Example: Slope=7 and y-intercept= -3

y=mx+b

y=7x-3

Question 10

What equation do you use to find the equation of a line when given slope and a point?

Answer 10

Substitute the slope and x and y points into y=mx+b, and solve for b (y-intercept)

Example: Slope= 3/4 Point= (-1,2)

y=mx+b

2=3/4 (-1) +b

2=-3/4+b

2/1+3/4=b

b= 8/4+ 3/4

b= 11/4

Question 11

What is the slope of 2 Parallel lines?

Answer 11

The slopes are equal.

Question 12

What is the slope of 2 Perpendicular lines?

Answer 12

They are the negative reciprocal of eachother (Flip the fraction)

Example: m= -1/2 and m=2/1

Question 13

What does No Solution look like?

Answer 13

Slope is the same, y-intercept is different

Question 14

What does One Solution look like?

Answer 14

Slopes are different

Question 15

What does Infinite Solutions look like?

Answer 15

Same slopes and same y-intercepts

Question 16

What does it mean to solve a linear system?

Answer 16

To find the point(s) of intersection of 2 equations

Question 17

What are the 5 methods to solving a linear system?

Answer 17

1) Table of values method
2) x and y-intercept method
3) y=mx+b method
4) Substitution
5) Elimination

Question 18

How do you check that your ordered pair is a solution to a linear system?

Answer 18

Substitute your ordered pair back into the equation. Both sides of the equation should be the same.

Question 19

How do you find the point of intersection by Substitution?

Answer 19

Example: y=3x-1 and 2x-y=9

Step 1) Label Equations 1 and 2

(1) y=3x-1
(2) 2x-y=9

Step 2) Rearrange one of the equations to isolate for x or y

y=3x-1

Step 3) Substiture the expressions from step 2 into the other equation and solve

Sub 1 into 2

2x- (3x-1)=9

2x-3x+1=9

• x-9-1
• x/-1=8/-1

x= -8

Step 4) Substitute the value found in step 3 into one of the original equations to find the other variable

Sub x= -8 into (1)

y=3x-1

y=3(-8) -1

y=-25

Step 5) Write a conclusion

Therefore the point of intersection is (-8, -25)

* Refer to “Solving Linear Systems by Substitution Method” for more information

Question 20

How do you find the point of intersection, by Elimination?

Answer 20

Example: x+3y=7 and x+y=3

Step 1) Label the equations 1 and 2

(1) x+3y=7
(2) x+y=3

Step 2) Multiply every term in one or both of the equations so that the variable you want to eliminate has the same or different coeffcient

x+y=3 → X -1 → -x-y= -3

Step 3) Add or subtract to eliminate one variable

0x+2y=4

Step 4) Solve for one variable

2y/2= 4/2

y=2

Step 5) Substitute the value found in step 4 into one of the original equations to find the the value of the other variable

Sub y=2 into (1)

x+3 (2) =7

x+6=7

x=7-6

x=1

Step 6) Write a Conclusion

Therefore the point of intersection is (1, -2)

*Refer to “Solving Liner Systems Using Elimination” for more information

Question 21

What is a First Difference Column?

Answer 21

A column on your table of values that tells the dfference between one number to the next. This helps you to Determine if the graph is linear or non-linear

Example: from 3 to 5 the first difference is 2

Question 22

What is a Second Difference Column?

Answer 22

A column on your table of values that tells the dfference between one number to the next, in the first difference column. This helps you to see if the graph is a parabola.

Question 23

How can you tell if a graph is linear or not by a table of values, and first difference column?

Answer 23

If the x and y columns increse or decrese by the same amount then the graph is linear.

Question 24

How can you tell if a graph is a parabola based on a table of values and a second difference column?

Answer 24

If the second difference columns increses or decreses by the same amount then the graph is a parabola.

Question 25

What is the equation of a vertical line, when given one point?

Answer 25

x= the x value in the point given

Example: (-3, 5)

Vertical Line Equation: x=-3

Question 26

What is an Opposite Reciprocal?

Answer 26

2 fractions with opposite numerators and denominators.

Example: -10/3 and -3/10