Multiple representations homework 7 answer key
Understanding how to approach the problems in this set, along with clear, step‑by‑step solutions, will help you master the concept of representing numbers and equations in various forms. Whether you’re tackling algebraic expressions, converting fractions, or graphing linear equations, this answer key breaks each problem into digestible parts and explains the reasoning behind every answer.
Introduction
The seventh homework set in the Multiple Representations unit focuses on translating between different mathematical forms—numerical, algebraic, graphical, and verbal. Mastery of these translations is essential for higher‑level math courses and real‑world problem solving. The key to success lies in recognizing patterns and applying systematic strategies That's the part that actually makes a difference..
Problem 1 – Converting Fractions to Decimals and Percentages
Answer:
- Fraction 7/8
- Decimal: ( \frac{7}{8} = 0.875 )
- Percentage: (0.875 \times 100 = 87.5%)
- Fraction 3/5
- Decimal: ( \frac{3}{5} = 0.6 )
- Percentage: (0.6 \times 100 = 60%)
Explanation:
Divide the numerator by the denominator to get the decimal. Multiply the decimal by 100 to obtain the percentage. Always keep two decimal places for fractions that don’t terminate neatly Practical, not theoretical..
Problem 2 – Converting Decimals to Fractions
Answer:
- 0.3 → ( \frac{3}{10} )
- 0.75 → ( \frac{75}{100} = \frac{3}{4} )
- 0.125 → ( \frac{125}{1000} = \frac{1}{8} )
Explanation:
Write the decimal over a power of ten matching the number of decimal places, then simplify by dividing numerator and denominator by the greatest common divisor.
Problem 3 – Writing Equations in Different Forms
Given: “The line passes through (2, 5) and has a slope of 3.”
Standard Form (Ax + By = C):
- Start with slope‑intercept: (y = 3x + b).
- Substitute point (2, 5): (5 = 3(2) + b \Rightarrow b = -1).
- Equation: (y = 3x - 1).
- Convert to standard form: (3x - y = 1).
Answer: (3x - y = 1)
Explanation:
Using the point–slope method gives a quick path to the slope‑intercept form, which is then rearranged into standard form.
Problem 4 – Sketched Graphs of Linear Equations
Equation: (y = -2x + 4)
Key Points:
- y‑intercept: (0, 4)
- x‑intercept: Set (y = 0): (0 = -2x + 4 \Rightarrow x = 2).
- Slope: (-2) (down 2 units for every right 1 unit).
Graphing Steps:
- Plot (0, 4).
- From (0, 4), move right 1, down 2 to (1, 2).
- Draw a straight line through these points, extending in both directions.
Answer: A line crossing the y‑axis at 4 and the x‑axis at 2, sloping downward.
Problem 5 – Solving Systems Using Multiple Representations
System:
[
\begin{cases}
2x + 3y = 12 \
-4x + y = 3
\end{cases}
]
Solution via Elimination:
- Multiply the second equation by 3: (-12x + 3y = 9).
- Add to the first equation: ((2x - 12x) + (3y + 3y) = 12 + 9).
- Simplify: (-10x + 6y = 21).
- Solve for (y) in terms of (x): (6y = 10x + 21 \Rightarrow y = \frac{10x + 21}{6}).
- Subtract this expression from the first equation or substitute into the second to find (x).
- Substituting into the second: (-4x + \frac{10x + 21}{6} = 3).
- Multiply by 6: (-24x + 10x + 21 = 18).
- Simplify: (-14x = -3 \Rightarrow x = \frac{3}{14}).
- Find (y): (y = \frac{10(\frac{3}{14}) + 21}{6} = \frac{\frac{30}{14} + 21}{6} = \frac{\frac{30 + 294}{14}}{6} = \frac{324/14}{6} = \frac{324}{84} = \frac{27}{7}).
Answer: (x = \frac{3}{14}), (y = \frac{27}{7})
Verification by Substitution:
Plugging back into the original equations confirms both are satisfied Worth knowing..
Problem 6 – Interpreting Word Problems into Equations
Word Problem: A store sells pens for $1.50 each and pencils for $0.75 each. If a customer buys a total of 20 items for $22.50, how many pens did they buy?
Equation Setup:
Let (p =) number of pens, (c =) number of pencils.
- (p + c = 20) (total items)
- (1.5p + 0.75c = 22.5) (total cost)
Solve:
From the first equation, (c = 20 - p). Substitute into the second:
(1.5p + 0.75(20 - p) = 22.5).
(1.5p + 15 - 0.75p = 22.5).
(0.75p = 7.5).
(p = 10).
Answer: The customer bought 10 pens (and consequently 10 pencils).
Problem 7 – Graphical Representation of Inequalities
Inequality: (y \le 2x + 1)
Graphing Steps:
- Sketch the boundary line (y = 2x + 1).
- Use a solid line because the inequality is “≤”.
- Shade the region below the line (including the line).
Key Points:
- Test point (0, 0): (0 \le 1) → true, so shade below.
- The shaded area represents all ordered pairs satisfying the inequality.
Problem 8 – Converting Between Exponential and Logarithmic Forms
Exponential: (2^x = 32)
Logarithmic Conversion:
(x = \log_2 32).
Calculating:
(32 = 2^5), so (x = 5).
Answer: (x = 5)
Explanation:
Recognizing the base and expressing the right‑hand side as a power of the base simplifies the solution.
Problem 9 – Checking for Consistency in a System
System:
[
\begin{cases}
x + y = 4 \
2x + 2y = 8
\end{cases}
]
Analysis:
Multiply the first equation by 2: (2x + 2y = 8).
The second equation is identical, so the system has infinitely many solutions along the line (x + y = 4) Most people skip this — try not to. But it adds up..
Answer: The system is consistent and dependent (infinitely many solutions).
Problem 10 – Word Problem Involving Ratios
Problem: The ratio of boys to girls in a class is 3:4. If there are 28 students, how many girls are there?
Setup:
Let (3k =) number of boys, (4k =) number of girls.
Total: (3k + 4k = 7k = 28).
(k = 4).
Girls: (4k = 16) Easy to understand, harder to ignore..
Answer: There are 16 girls in the class.
FAQ
Q1: How do I remember which form to use for a given problem?
A1: Ask yourself what the problem asks: a numerical answer, an equation, a graph, or a verbal description. Match the desired output to the appropriate representation.
Q2: What if a decimal repeats?
A2: Use a fraction with a prime denominator. Here's one way to look at it: (0.\overline{3} = \frac{1}{3}).
Q3: How can I quickly sketch a line from its equation?
A3: Identify the y‑intercept (set (x=0)). Find the x‑intercept (set (y=0)). Plot both points, then draw the line.
Conclusion
Mastering multiple representations equips you to handle any mathematical landscape—from simple fraction conversions to complex systems of equations. By consistently practicing the steps highlighted above, you’ll build confidence and precision in translating numbers, equations, and graphs into one another. Keep the key strategies in mind, and soon solving these problems will become a natural, intuitive process.
