Envision Math Grade 8 Volume 1 Chapter 2: Analyze And Solve Linear Equations
Page 147 Question 1 Answer
We need to explain how we can analyze connections between linear equations and use them to solve problems.
Equations that consist of degree one are said to be linear equations.
Linear equations will always result in a straight line when plotted on the graph.
The highest degree of the variables present in the linear equations must be one.
A linear equation must have a constant in it.
Linear equations can be solved by doing arithmetical operations on both sides of the equation. This will not affect the balance of the equation.
It can also be solved graphically.
Some of the examples of linear equations are,
5x + 2y = 3
\(2 y=\frac{15}{2}\)15x – 5 = 0
We can analyze connections between linear equations by solving them arithmetically or by graphically, and we can also use them to solve real-life problems.
Page 147 Exercise 3 Answer
The slope intercept form of a line is y = mx + b where them is the slope of the line and the b is the y−intercept of the line.
Example:
If we look at the graph of given line we can see the equation y = −2x + 3.
The slope of the given line is −2 and that means that our graph of the line will be decreasing. The y− intercept is 3 which tells us where the line will cross the y−axis.
The slope intercept form of a line is y = mx + b. The variable m in the equation stands for the slope. The variable bin the equation stands for the y−intercept.
Page 147 Exercise 1 Answer
Given that, Paddleboats rent for a fee of $25, plus an additional $12 per hour. We need to write the equation, in y = mx + b form, represents the cost to rent a paddleboat for x hours. Also, we have to explain how you write the equation. Use vocabulary words in your explanation.
The equation of the line is of the form y=mx+b
Here, b is a constant.
x is the number of hours while y is the cost to rent a paddleboat.
Given that, the initial investment is $25
The additional cost is $12 per hour.
Thus, the equation will be
y = 12x + 25
The equation is y = 12x + 25. Here, y is the cost to rent a paddleboat where x is the number of hours.
Page 148 Exercise 1 Answer
The given equation is 2x + 6x = 1000
We need to solve the given equation and find the value of x
The value of x = 125
Page 148 Exercise 2 Answer
The given equation is \(2 \frac{1}{4} x+\frac{1}{2} x=44\)
We need to solve the given equation and find the value of x
The value of x = 16
Page 148 Exercise 3 Answer
The given equation is −2.3x − 4.2x = −66.3
We need to solve the given equation and find the value of x
The value of x = 10.2
Page 148 Exercise 4 Answer
Given that, Javier bought a microwave for $105. The cost was 30% off the original price. We need to find the price of the microwave before the sale.
Let x be the price of the microwave before the sale.
The price of the microwave before the sale is $150
Page 148 Exercise 3 Answer
The given equation is 9x − 5x + 18 = 2x + 34
We need to solve the given equation and find the value of x
The value of x = 8
Page 149 Exercise 1 Answer
The given equation is 4(x+4) + 2x = 52
We need to solve the given equation and find the value of x
The value of x = 6
Page 149 Exercise 2 Answer
The given equation is 8(2x+3x+2) = −4x + 148
We need to solve the given equation and find the value of x
The value of x = 3
Page 149 Exercise 3 Answer
Given that, Justin bought a calculator and a binder that were both 15% off the original price. The original price of the binder was $6.20. Justin spent a total of $107.27. We need to find the original price of the calculator.
The price of the binder at which Justin bought is,
\(6.20 \times \frac{100-15}{100}=6.20 \times \frac{85}{100}\)= \(\frac{527}{100}\)
= 5.27
The total price spent by Justin is,
5.27 + x = 107.27
x = 102
This x be the discounted price of the calculator.
Therefore, the calculator’s original price will be,
The original price of the calculator is $120
Page 149 Exercise 1 Answer
The given equation is x + 5.5 + 8 = 5x − 13.5 − 4x
We need to solve the given equation and find the value of x
The equation has no solutions.
The given equation doesn’t have any solutions.
Page 149 Exercise 2 Answer
The given equation is \(4\left(\frac{1}{2} x+3\right)=3 x+12-x\)
We need to solve the given equation and find the value of x
The given equation doesn’t have any solutions.
Page 149 Exercise 4 Answer
Given that, the weight of Abe’s dog can be found using the expression 2(x+3), where x is the number of weeks. The weight of Karen’s dog can be found using the expression 3(x+1), where x is the number of weeks. We need to determine when will the dogs ever be the same weight.
Equating both the expressions, we get,
The dogs be the same weight after 3 weeks.
Page 150 Exercise 2 Answer
Given that, A 16-ounce bottle of water from Store A costs $1.28. The cost in dollars, y, of a bottle of water from Store B is represented by the equation y = 0.07x, where x is the number of ounces. We need to find the cost per ounce of water at each store. Also, find which store’s bottle of water costs less per ounce.
Cost of one ounce of water at store A,
= \(\frac{1.28}{16}\)= 0.08 dollars per ounce
Cost of one ounce of water at store B,
y = 0.07x
y = 0.07(1)
y = 0.07 dollars per ounce
Therefore, store B’s bottle of water costs less per ounce.
Store B’s bottle of water costs less per ounce.
Page 151 Exercise 2 Answer
Given that, A mixture of nuts contains 1 cup of walnuts for every 3 cups of peanuts.
We need to graph the line.
The linear equation formed from the given data is
y = 3x
Graphing the given equation, we get,
The graph is,
Page 152 Exercise 1 Answer
We need to graph the line with the equation \(y=\frac{1}{2} x-1\)
Finding two points to draw the line,
Plot both the points (0, -1) and (2, 0) on the graph and connect them together with the line.
The graph will be,
The graph of the equation is,
Page 153 Exercise 1 Answer
Given that, each block below shows an equation and a possible solution. We need to shade a path from START to FINISH. Also, follow the equations that are solved correctly. You can only move up, down, right, or left.
Substituting each value in its corresponding equation, we get,
x = 2 ⇒ 2x + 3 = 7
2(2) + 3 = 7
4 + 3 = 7
7 = 7
TRUE
y = −1 ⇒ 9(−1)−1 = −10
−10 = −10
TRUE
t = 2 ⇒ 5(2) + 1 = 9
11 ≠ 9
FALSE
x = −1 ⇒ −11(−1) + 12 = 1
23 ≠ 1
FALSE
Repeat the same in the second row, we get,
p = −7 ⇒ 19−4(−7) = 9
19 + 28 = 9
47 ≠ 9
FALSE
j = 60 ⇒ 30−60 = 90
−30 ≠ 90
FALSE
m = 7 ⇒ 14 + 3(7) = 35
35 = 35
TRUE
h = 4 ⇒ 6(4)−1 = 25
23 ≠ 25
FALSE
Repeat the same in the third row, we get,
t = 5 ⇒ 20(5)−1 = 95
99 ≠ 95
FALSE
q = 3 ⇒ 20−3 = 17
17 = 17
TRUE
w = −1 ⇒ −4(−1) + 7 = 11
11 = 11
TRUE
a = 2 ⇒ −2 + 15 = 13
13 = 13
TRUE
Repeat the same in the fourth row, we get,
y = 4 ⇒ 7(4) + 4 = 32
32 = 32
TRUE
y = 6 ⇒ 23 = 1 + 4(6)
23 ≠ 25
FALSE
r = −9 ⇒ −9(−9)−4 = −85
77 ≠ −85
FALSE
x = −25 ⇒ 100 − 4(−25) = 0
200 ≠ 0
FALSE
Repeat the same in the fifth row, we get,
b = −4 ⇒ −6(−4) + 27 = 3
51 ≠ 3
FALSE
\(z=\frac{1}{2} \Rightarrow 2\left(\frac{1}{2}\right)+1=0\)FALSE
x = −1 ⇒ 47−2(−1) = 45
49 ≠ 45
FALSE
k = 6 ⇒ −12 + 9(6) = 42
42 = 42
TRUE
The correct ones are marked as “T” while the incorrect ones are marked as “F”.
