1.
Given the following equations, show \(x=y\text{.}\) For each step in the process, state the property of equality you are using.
\(x=2a+7-a \ y=3a-8-2a+15 \)
Section 2.4 Supplementary Exercises
Subsection 2.4.1 Equality Questions
Exercises Exercises
2.
Is the equality true? Explain.
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\(\displaystyle 2^0 = 63^0\)
3.
For the next two questions, consider the following relationships: let \(x=2\text{,}\) \(y=x+b\text{,}\) \(z=c-y+a\text{.}\)
4.
In the following process, label each step with the property of equality used to reach it.
\begin{align*}
d \amp = x - y + z\\
\amp = 2 - (x + b) + c - y + a\\
\amp = 2 - x - b + c - y + a\\
\amp = 2 - 2 - b + c - (x + b) + a\\
\amp = -b + c - x - b + a\\
\amp = a - 2b + c - x\\
\amp = a - 2b + c - 2
\end{align*}
5.
6.
Prove that all three properties of equality hold for the biconditional statement "\(A\) if and only if \(B\text{.}\)"
7.
Prove that set equality is reflexive, transitive, and symmetric.
8.
Which properties (of reflexive, transitive, symmetric) hold for the following relations? Do they always hold, never hold, or only hold for some inputs?
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\(\displaystyle <\)
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\(\displaystyle >\)
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\(\displaystyle \leq\)
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\(\displaystyle \geq\)
9.
Formulate a relationship that exhibits the given property, but neither of the other two.
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Reflexivity.
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Transitivity.
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Symmetry.
10.
Determine whether each set equality is true or not.
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\(\displaystyle \{a/b ∣ a,b ∈ ℤ, a<b\} = \{x ∈ ℚ ∣ x<1, \; |x|<1\}\)
-
\(\displaystyle \{1,2,3\} ∪ \{3,4\} = \{1,2,3,4\}\)
-
\(\displaystyle (\{1,2,3\} ∩ \{2,3,4\}) ∪ \{5\} = \{2,3,5\}\)
-
\(\displaystyle \{x ∈ ℤ ∣ x \text{ is even}\} = \{2n ∣ n ∈ ℤ\}\)
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\(\displaystyle \{x ∈ ℝ ∣ x^2 < 4\} = \{-2,2\}\)
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\(\displaystyle \{x ∈ ℤ ∣ x^2 \leq 9\} = \{-3,-2,-1,0,1,2,3\}\)
-
\(\displaystyle \{x ∈ ℕ ∣ x \text{ is prime and \} x<10} = \{2,3,5,7\}\)
Subsection 2.4.2 Problem Solving Questions
Example 2.4.1. Solving a system by graphing.
One way to solve systems of equations is to graph each equation on the same axes; their intersection point(s) are common solutions to the system. The following example illustrates this approach.
Example 1. Consider the system \(5x + 7y = 69\text{,}\) \(9x - 4y = 8\text{.}\) We first solve each equation for the same variable (here, \(y\)) to make graphing straightforward.
\begin{align*}
5x + 7y \amp = 69\\
7y \amp = -5x + 69\\
y \amp = -\dfrac{5}{7}x + \dfrac{69}{7}
\end{align*}
\begin{align*}
9x - 4y \amp = 8\\
-4y \amp = -9x + 8\\
y \amp = \dfrac{9}{4}x - 2
\end{align*}
From the graph, the lines intersect at \((4,7)\text{,}\) so the solution is \(x=4\text{,}\) \(y=7\text{.}\) You can verify by substitution into the original equations. The choice to solve for \(y\) was arbitrary—solving for \(x\) would lead to the same intersection point.
Conceptually, solving a system means finding values that satisfy all equations simultaneously. Points on a graph represent solutions to their respective equations; intersections represent common solutions. While graphing gives geometric insight, algebraic methods (substitution or elimination) are usually faster, especially for systems with more than two variables.
Exercises Exercises
1.
\begin{align*}
x+4-2(3x-5) \amp = \dfrac{x+4}{3}\\
axy+bx+4+7y \amp = 46\\
3 \amp = 24 - c\\
x^2 + 33x + 33 \amp = 1\\
3 \amp = 4 - 1
\end{align*}
2.
Identify the terms in each equation.
3.
For each term involving a variable, identify the variable(s) and coefficient.
4.
For each equation, identify the constant(s).
5.
For each equation, determine whether it is linear or not. Justify briefly.
6.
Define any necessary variables and write an equation to represent each word problem.
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Henry started with 7 barrels of maple syrup and receives 4 more every week. How many weeks until he has 63 barrels?
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Zach spent \$219.80 on frozen pizzas; each costs \$7.85. How many pizzas did he buy?
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Harris buys Rory’s (380 g, 47 pretzels per bag) and Pete’s (400 g, 39 pretzels per bag). He has 493 pretzels totaling 4240 g. How many bags of each?
7.
Blueberry muffins cost \$2 and chocolate chip muffins cost \$3 at Jolene’s bakery. Chris spent \$27 buying only these muffins. How many different ordered pairs \((x,y)\) of blueberry/chocolate chip muffin counts are possible? How do you know you found them all?
8.
For each equation/system, write a word problem that would be solved by it.
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\(\displaystyle x + 36 = 180\)
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\(\displaystyle 3x + 50 = 200\)
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\(18y - 3x = 90\text{,}\) \(10y + 5x = 72\)
9.
Solve each system by elimination.
-
\(2x+y=47\text{,}\) \(2x-3y=19\)
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\(10x-3y+20=35\text{,}\) \(14y-8x=2\)
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\(13x+2y=228\text{,}\) \(2y-3x=-44\)
-
\(3x=149-10y\text{,}\) \(11y=142+4x\)
10.
By which process is the following system being solved? How do you know?
\(7x-2y=19,\; 3x+4y=149\)
\begin{align*}
7x-2y \amp = 19\\
7x \amp = 19 + 2y\\
x \amp = \dfrac{19}{7} + \dfrac{2}{7}y
\end{align*}
\begin{align*}
3\!\left(\dfrac{19}{7} + \dfrac{2}{7}y\right) + 4y \amp = 149\\
\dfrac{57}{7} + \dfrac{6}{7}y + 4y \amp = 149\\
57 + 6y + 28y \amp = 1043\\
34y \amp = 986\\
y \amp = 29
\end{align*}
11.
Determine whether each equation/system can be solved over the reals. If so, describe the solution set’s form (unique, none, or infinitely many). Explain how you know.
-
\(3x + 2y = 20\text{,}\) \(8x + 7y = 65\)
-
\(x - 7y + 74 = 18z\text{,}\) \(10x - 32y = 120\)
-
\(3xy + 12y = 10\text{,}\) \(12x - 3y = 90\)
12.
Jonah solved \(19y=7x+27\text{,}\) \(3x-4y=5\) and found \(x=-\dfrac{2437}{2040}\text{,}\) \(y=\dfrac{116}{85}\text{.}\) How could he check his answer? Check it, and if incorrect, find the correct solution.
13.
Identify the errors (if any) in each solution and find the correct solution.
-
\begin{align*} 2y + 20 \amp = 14x\\ 2y \amp = 14x - 20\\ y \amp = 7x - 10 \quad \text{(watch the division by 2!)}\\ 4x + 3(7x-10) \amp = 45 \end{align*}
-
\begin{align*} 37 - z - 4w \amp = 0\\ z \amp = 37 - 4w \quad \text{(isolate correctly)}\\ 3(37 - 4w) - 6w \amp = 3\\ 111 - 12w - 6w \amp = 3\\ 111 - 18w \amp = 3 \end{align*}
14.
Subsection 2.4.3 Explanatory and Critical Thinking Questions
Exercises Exercises
1.
A student says: “Graphing is always the best way to solve systems of equations, because you can see the answer directly.” Do you agree or disagree? Explain when graphing is helpful, and when other methods (substitution, elimination) are more practical.
2.
Another student claims: “If two equations look different, then they must have different solution sets.” Give a counterexample, and explain why equations can look different but still represent the same line or same set of solutions.
3.
Consider the system \(3x + 2y = 6\text{,}\) \(6x + 4y = 12\text{.}\) A student concludes that there is exactly one solution because the equations are not identical. Explain the flaw in this reasoning, and describe the actual solution set.
4.
A student is solving by elimination and subtracts one equation from another incorrectly, losing a variable in the process. How can you recognize when an algebraic error has happened, and how does checking with substitution into the original system help?
5.
Explain the difference between “an equation has no solution,” “a system has infinitely many solutions,” and “a system has exactly one solution.” Give an example of each, and explain how you would recognize it graphically and algebraically.
6.
Consider the equation \(2x + 5 = 2x + 7\text{.}\) A student says: “We can solve this and find \(x\text{.}\)” Explain why this equation cannot be solved for a single \(x\text{,}\) and describe what it means for a system or equation to be inconsistent.
7.
Word problems require translating between language and algebra. Take the sentence: “The sum of two numbers is 30, and one number is 4 greater than the other.” Explain how to define variables, write equations, and check that the system models the words correctly.
