Chapter 4 Represent And Solve Equations And Inequalities
Midpoint
Page 209 Exercise 1 Answer
To solve equations, we isolate the variable using inverse relationships and the properties of equality. To solve addition equations, we use the Subtraction Property of Equality, and to solve subtraction equations, we use the Addition Property of Equality since addition and subtraction are inverse operations. To solve multiplication equations, we use the Division Property of Equality, and to solve division equations, we use the Multiplication Property of Equality since multiplication and division are inverse operations.
Page 209 Exercise 2 Answer
The equation is f × 4 = \(\frac{1}{2}\). To solve the equation either divide both sides by four or multiply both sides by the reciprocal of 4, that is by \(\frac{1}{4}\).
f x 4 = \(\frac{1}{2}\)
\(f \times 4 \times \frac{1}{4}=\frac{1}{2} \times \frac{1}{4}\) (Multiply both sides by \(\frac{1}{4}\).)
f = \(\frac{1}{2} \times \frac{1}{4}\)
f = \(\frac{1 \times 1}{2 \times 4}\)
f = \(\frac{1}{8}\) (Multiply the fractions.)
Result
f = \(\frac{1}{8}\)
Read And Learn More: enVisionmath 2.0 Grade 6 Volume 1 Solutions
Page 209 Exercise 3 Answer
The equations is 832 ÷ n = 16. Start by multiplying by n.
832 ÷ n = 16
832 ÷ n x n = 16 x n (Multiply both sides by n.)
832 = 16 x n (Simplify.)
832 ÷ 16 = 16 × n ÷ 16 (Divide both sides by 16.)
52 = n
Result
n = 52
Page 209 Exercise 4 Answer
The equation is x − 10 = 6. To solve start by adding 10 to both sides.
x − 10 = 6
x – 10 + 10 = 6 + 10 (Add 10 to both sides.)
x = 16 (Evaluate.)
Result
x = 16
Page 209 Exercise 5 Answer
n – n – 9 = 12 – n (The Subtraction Property of Equality was used.)
n – 9 + 12 = 12 – 9 (The equations are not equivalent.)
n – 9 + 9 = 12 + 9 (The Addition Property of Equality was used.)
n – 9 – n = 12 – n (The Subtraction Property of Equality was used.)
n – 9 + 9 = 12 – 12 (The equations are not equivalent..)
The equations which are not equivalent to n − 9 = 12 are: n − 9 + 9 = 12 − 12 and n − 9 + 12 = 12 − 9.
Result
n − 9 + 9 = 12 − 12, n − 9 + 12 = 12 − 9.
Page 209 Exercise 6 Answer
To answer the question use substitution – substitute d for each of the values and check it the equation is true.
9 = 18 ÷ d
9 = 18 ÷ 0.5
9 = 180 ÷ 5 (Multiply both numbers by 10)
9 ≠ 36
9 = 18 ÷ \(\frac{10}{5}\)
9 = 18 ÷ 2 (Simplify)
9 = 9
9 = 18 ÷ 162
9 = \(\frac{18}{162}\)
9 ≠ \(\frac{1}{9}\)
9 = 18 ÷ \(\frac{1}{4}\)
9 = 18 x 4 (Multiply by the reciprocal of \(\frac{1}{4}\))
9 ≠ 72
The only value of d that made the equation 9 = 18 ÷ d true was \(\frac{10}{5}\).
Result
\(\frac{10}{5}\)Page 209 Exercise 7 Answer
Substitute the variables with their values in the expression A = bh. Solve the equation to find the height.
A = bh
15.3 = bh
15.3 = 4.5 h (Substitute variables with their values.)
15.3 ÷ 4.5 = h (Divide both sides by 4.5.)
3.4 = h
The height of the parallelogram is 3.4 centimeters.
Result
3.4 centimeters.
Page 210 Exercise 1a Answer
To solve the equation \(\frac{3}{4}\)h = \(3 \frac{3}{8}\) using a reciprocal, multiply both sides of the equation by the reciprocal of \(\frac{3}{4}\) which is \(\frac{4}{3}\).
The equation than look like:
\(\frac{3}{4} h \times \frac{4}{3}=3 \frac{3}{8} \times \frac{4}{3}\)Result
\(\frac{3}{4} h \times \frac{4}{3}=3 \frac{3}{8} \times \frac{4}{3}\)Page 210 Exercise 1b Answer
From Part A, we know the equation \(\frac{3}{4} h=3 \frac{3}{8}\) can be used to find the height h of the finished totem. In Part A, we determined that the equation \(\frac{3}{4} h \times \frac{4}{3}=3 \frac{3}{8} \times \frac{4}{3}\) was an equivalent equation. Simplifying this equivalent equation gives:
\(\frac{3}{4} h \times \frac{4}{3}=3 \frac{3}{8} \times \frac{4}{3}\)
h = \(3 \frac{3}{8} \times \frac{4}{3}\) Simplify the left side.
h = \(\frac{27}{8} \times \frac{4}{3}\) Rewrite the mixed number as an improper fraction.
h = \(\frac{108}{24}\) Multiply.
h = \(\frac{9}{2}\) Reduce the fraction.
h = \(4 \frac{1}{2}\) Rewrite as a mixed number.
The height of the finished totel pole is then \(4 \frac{1}{2}\) ft.
Next, we need to write and solve an equation to find the height, s, of the section that has not been carved. We know that \(3 \frac{3]{8}\) ft of the totem has been completed so the finished totem pole will have a height of s + \(3 \frac{3}{8}\) feet. Since the height of the finished totem pole is \(4 \frac{1}{2}\) ft, then the equation is \(s+3 \frac{3}{8}=4 \frac{1}{2}\).
Solving this equation for s we get:
\(s+3 \frac{3}{8}=4 \frac{1}{2}\)
\(s+3 \frac{3}{8}-3 \frac{3}{8}=4 \frac{1}{2}-3 \frac{3}{8}\) Subtract \(3 \frac{3}{8}\) on both sides.
s = \(4 \frac{4}{8}-3 \frac{3}{8}\) Get a common denominator.
s = \(1 \frac{1}{8}\) Subtract.
The height of the section that has not been carved is then \(1 \frac{1}{8}\)ft.
Result
\(h=4 \frac{1}{2} \mathrm{ft} \quad s+3 \frac{3}{8}=4 \frac{1}{2} \quad s=1 \frac{1}{8} \mathrm{ft}\)Page 210 Exercise 1c Answer
To answer the question solve the equation $10.50 + x = $19.35 Start by subtracting 10.50 from both sides.
10.50 + x − 10.50 = 19.35 − 10.50
x = 8.85
The cost of the wood Ronald used is $8.85.
Result
$8.85
Page 210 Exercise 1d Answer
To solve the equation $10.50 + y = $35.19 Ronald could use the Subtraction Property of Equality, that is he could subtract 10.50 from both sides. If he does so the unknown y stays on one side of the equation, since he used a property of equality the equation stayed balanced and the other side of the equation than shows the value of y.
$10.50 + y = $35.19
10.50 + y − 10.50 = 35.19 − 10.50
y = $24.69
Result
y = $24.69