Keystone Remediation Packet Week 2
Linear Equations
Assigned: Friday, April 12
Due: Friday, April 19
You must show all work to receive full credit
Linear Equation: A linear equation is an algebraic equation in which each term is either a constant or a
product of a constant and a single variable to the 1st power.
Equations of Lines:
Slope:
𝑦2 −𝑦1
𝑥2 −𝑥1
Slope Intercept Form:
y = mx + b
Standard Form:
ax + by = c
Point-Slope Form:
y - y1 = m(x – x1)
Graphing a linear equation: Any line is defined by two points, therefore, in order to graph a linear equation,
you must first plot two points. We find these two points based on the form of the linear equation (see above).
y
(x2, y2)
(a, 0)
x
(y2 - y1)
(0, b)
x2 - x1
Intercepts: In order to find the x-intercept(s) of an equation, you have to set y equal to zero and solve the
equation for x. In order to find the y-intercepts of an equation, you have to set x equal to zero and solve the
equation for y.
Parallel Lines: Two lines whose graphs have the same slope.
Perpendicular Lines: Two lines whose graphs have opposite reciprocal slopes.
Solving Linear Equations: In order to solve a linear equation, you must use reverse operations to isolate the
variable.
3𝑥 − 4 = 8
+4
+4
3𝑥 = 12
÷3
÷3
𝑥=4
System of Linear Equations: A system of linear equations is a group of n linear equations containing n
variables. The three most common ways to solve a system of linear equations are graphing, substitution and
elimination.
𝑦 = 2𝑥 − 3
𝑦 =𝑥−1
Solve by substitution:
Solve by elimination:
1. Isolate a single variable.
1. Write each equation in standard form.
2. Substitute into the other equation.
2. Multiply one/both equation by a constant
3. Solve for the first variable.
to allow one of the variable terms to cancel.
4. Substitute into the original equation.
3. Add the two equations together.
5. Solve for the second variable.
4. Solve for the first variable.
𝑦 = 2𝑥 − 3
𝑦 = 2(2) − 3
5. Substitute into an original equation.
𝑦 =𝑥−1
𝑦 =4−3
6. Solve for the second variable.
𝑥 − 1 = 2𝑥 − 3
𝑦=1
2𝑥 − 𝑦 = 3
2𝑥 − 𝑦 = 3
−2(𝑥 − 𝑦 = 1) 
−2𝑥 + 2𝑦 = −2
−𝑥 − 1 = −3
−𝑥 = −2
𝑥=2
0𝑥 + 𝑦 = 1
1=𝑥−1
𝑥=2

𝑦=1
Solve by graphing:
1. Graph each of the lines.
2. Find the intersection point.
y
5
4
3
2
1
–5
–4
–3
–2
–1
–1
(2, 1)
1
2
3
4
5
x
–2
–3
–4
–5
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1. You are driving to visit a friend in another state who lives 440 miles away. You are driving 55 miles per hour
and have already driven 275 miles. Write and solve an equation to find how much longer in hours you must
drive to reach your destination.
a.
b.
____
;
;
;
;
2. A customer went to a garden shop and bought some potting soil for $17.50 and 4 shrubs. The total bill was
$53.50. Write and solve an equation to find the price of each shrub.
a. 4p + $17.50 = $53.50; p = $9.00
b. 4(p + $17.50) = $53.50; p = $4.00
____
c.
d.
c. 4p + 17.5p = $53.50; p = $2.49
d. 4p + $17.50 = $53.50; p = $11.25
3. John and 2 friends are going out for pizza for lunch. They split one pizza and 3 large drinks. The pizza cost
$14.00. After using a $7.00 gift certificate, they spend a total of $12.10. Write an equation to model this
situation, and find the cost of one large drink.
a.
b.
c.
d.
3d + $14.00 – $7.00 = $12.10; $1.70
2d + $14.00 – $7.00 = $12.10; $2.55
3d – $14.00 + $7.00 = $12.10; $9.55
3d + $14.00 – $7.00 = $12.10; $1.90
____
4. Jenny has a job that pays her $8 per hour plus tips (t). Jenny worked for 4 hours on Monday and made $65 in
all. Which equation could be used to find t, the amount Jenny made in tips?
a. 65 = 4t + 8
b. 65 = 8t  4
c. 65 = 8t + 4
d. 65 = 8(4) + t
Find the x- and y-intercept of the line.
____
5. 2x + 3y = –18
a. x-intercept is 18; y-intercept is 18.
b. x-intercept is –6; y-intercept is –9.
c. x-intercept is 2; y-intercept is 3.
d. x-intercept is –9; y-intercept is –6.
Write the slope-intercept form of the equation for the line.
____
6.
y
5
4
3
2
1
–5
–4
–3
–2
–1
–1
1
2
3
4
5
x
–2
–3
–4
–5
a. y = 3 x  1
b. y = 3 x  1
____
c.
1
x 1
3
d.
1
y= x 1
3
y=
7. A pizza restaurant charges for pizzas and adds a delivery fee. The cost (c), in dollars, to have any number of
pizzas (p) delivered to a home is described by the function c = 8p + 3. Which statement is true?
a.
b.
c.
d.
The cost of 8 pizzas is $11.
The cost of 3 pizzas is $14.
Each pizza costs $8 and the delivery fee is $3.
Each pizza costs $3 and the delivery fee is $8.
____
8. Use the slope and y-intercept to graph the equation.
y=
3
x–3
4
y
a.
–5
–4
–3
–2
5
5
4
4
3
3
2
2
1
1
–1
–1
1
2
3
4
5
x
–3
–2
–4
–3
–2
–1
–1
–2
–3
–3
–4
–4
–5
–5
y
–4
–5
–2
b.
–5
y
c.
5
4
4
3
3
2
2
1
1
1
2
3
4
5
x
2
3
4
5
x
1
2
3
4
5
x
y
d.
5
–1
–1
1
–5
–4
–3
–2
–1
–1
–2
–2
–3
–3
–4
–4
–5
–5
Match the equation with its graph.
____
9. –7x + 7y = –49
y
a.
–10 –8
–6
–4
10
10
8
8
6
6
4
4
2
2
–2
–2
2
4
6
8
10
x
–4
–6
–4
–2
–2
–4
–6
–6
–8
–8
–10
–10
y
–6
–10 –8
–4
b.
–10 –8
y
c.
10
8
8
6
6
4
4
2
2
2
4
6
8
10
x
4
6
8
10
x
2
4
6
8
10
x
y
d.
10
–2
–2
2
–10 –8
–6
–4
–2
–2
–4
–4
–6
–6
–8
–8
–10
–10
Graph the equation.
____ 10. y + 2 = –(x – 4)
y
a.
–10 –8
–6
–4
10
5
8
4
6
3
4
2
2
1
–2
–2
2
4
6
8
10
x
–4
–4
–3
–2
–1
–1
–2
–6
–3
–8
–4
–10
–5
y
–6
–5
–4
b.
–10 –8
y
c.
5
8
4
6
3
4
2
2
1
2
4
6
8
10
x
2
3
4
5
x
1
2
3
4
5
x
y
d.
10
–2
–2
1
–5
–4
–3
–2
–1
–1
–4
–2
–6
–3
–8
–4
–10
–5
Is the relationship shown by the data linear? If so, model the data with an equation.
____ 11.
x
y
–9
–2
–5
–7
–1
–12
–17
3
a.
4
(x + 9).
5
b.
4
The relationship is linear; y + 9 =  (x + 2).
5
c. The relationship is not linear.
d.
5
The relationship is linear; y + 2 =  (x + 9).
4
The relationship is linear; y + 2 =
____ 12. What is the solution of the system of equations?
a. (–8, –15)
b. (–2, –15)
c. (0, 1)
d. (2, 5)
c. (4, 19)
d. (–8, –17)
c. (–3, 1)
d. (0, 4)
____ 13. What is the solution of the system of equations?
y = 3x + 7
y=x–9
a. (–1, –10)
b. (–17, –8)
____ 14. What is the solution of the system of equations?
2x – 2y = –8
x + 2y = –1
a. (–14, 1)
b. (1, 5)
____ 15. Which graph represents the following system of equations?
y = –2x + 1
y=x–2
y
a.
–4
–2
4
4
2
2
O
2
4
x
–2
O
–2
–4
–4
y
–2
–4
–2
b.
–4
y
c.
4
2
2
2
4
x
4
x
2
4
x
y
d.
4
O
2
–4
–2
O
–2
–2
–4
–4
____ 16. One of the steps Jamie used to solve an equation is shown below.
-5(3x + 7) = 10
-15x + -35 = 10
Which statements describe the procedure Jamie used in this step and identify the property that justifies the
procedure?
a. Jamie added -5 and 3x to eliminate the parentheses. This procedure is justified by the
associative property.
b. Jamie added -5 and 3x to eliminate the parentheses. This procedure is justified by the
distributive property.
c. Jamie multiplied 3x and 7 by -5 to eliminate the parentheses. The procedure is justified by
the distributive property.
d. Jamie multiplied 3x and 7 by -5 to eliminate the parentheses. The procedure is justified by
the associative property.
____ 17. Francisco purchased x hot dogs and y hamburgers at a baseball game. He spent a total of $10. The equation
below describes the relationship between the number of hot dogs and the number of hamburgers purchased.
3x + 4y = 10
The ordered pair (2,1) is the solution to the equation. What does the solution represent?
a.
b.
c.
d.
Hamburgers cost 2 times as much as hot dogs.
Francisco purchased 2 hot dogs and 1 hamburger.
Hot dogs cost $2 each and hamburgers cost $1 each.
Francisco spent $2 on hot dogs and $1 on a hamburger.
____ 18. Samantha and Maria purchased flowers. Samantha purchased 5 roses for x dollars each and 4 daisies for y
dollars each and spent $32 on the flowers. Maria purchased 1 rose for x dollars and 6 daisies for y dollars each
and spent $22. The system of equations is represented below.
5x + 4y = 32
x + 6y = 22
Which statement is true?
a.
b.
c.
d.
A rose costs $1 more than a daisy.
Samantha spent $4 on each daisy.
Samantha spent more on daisies than she did on roses.
Samantha spent over 4 times as much on daisies as she did on roses.
____ 19. Jeff’s restaurant sells hamburgers. The amount charged for a hamburger (h) is based on the cost for a plain
hamburger plus an additional charge for each topping (t) as shown in the equation below.
h = 0.06t + 5
What does the number 0.06 represent in the equation?
a.
b.
c.
d.
the number of toppings
the cost of a plain hamburger
the additional cost of each topping
the cost of a hamburger with 1 topping
____ 20. A juice machine dispenses the same amount of juice into a cup each time the machine is used. The equation
below describes the relationship between the number of cups (x) into which juice is dispensed and the gallons of
juice (y) remaining in the machine.
x + 12y = 180
How many gallons of juice are in the machine when it is full?
a. 12
b. 15
c. 168
d. 180