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4.2 — Elvira’s Equations
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Rearrange a formula to find a different quantity, and use units to understand what an expression means.
1. In class: A Solidify Understanding Task
Solving for a variable in a formula
A formula is an equation describing a relationship between quantities. A literal equation contains more than one variable. To solve for a particular variable, rearrange the equation so that variable is alone on one side. The answer will usually be an expression containing the other variables.
Use the same operations as in Section 4.1: do the same thing to both sides to keep them equal. Letters stand for numbers, so you can add or subtract a variable, or multiply or divide by it. A quantity you divide by must be nonzero.
What do the quantities measure?
Elvira’s variables describe lunch counts, prices, and serving time. Use these meanings throughout the main task:
| Symbol | Meaning | Units |
|---|---|---|
| \(S\), \(W\), \(P\) | Students buying salad, sandwich, and pizza lunches, respectively | Students |
| \(F\) | Food servers | Servers |
| \(C\) | Classes in the school | Classes |
| \(M_T\) | Time to serve all students | Minutes |
| \(P_L\) | Price per lunch | Dollars per lunch |
The subscript distinguishes a variable: \(P_L\) is the price per lunch, while \(P\) counts pizza customers. \(P_L\) is one variable, not \(P\) multiplied by \(L\).
Units help you interpret an expression. Add or subtract quantities measured in the same units. Dividing a student count by a class count gives students per class. Multiplying dollars per lunch by the number of lunches gives dollars.
Worked example 1 — Recover the salad-line count
Original formula — Main 1–3: \(A=\frac{S+W+P}{C}\)
\(A\) is the average number of lunch-buying students per class. To solve for \(S\), multiply both sides by the denominator \(C\) first, then subtract \(W\) and \(P\):
This makes sense: \(AC\) is the total number of students buying lunch. Removing the sandwich and pizza customers leaves the salad customers.
Worked example 2 — Find the price per lunch
Original formula — Main 4–6: \(R=P_L(S+W+P)\)
\(R\) is total lunch revenue, measured in dollars. Divide both sides by the total number of lunches:
The result is dollars per lunch: total revenue divided by the number of lunches sold.
In Main 7, \(T=S+W+P\) names the total number of students buying lunch. It lets us write the revenue equation more simply as \(R=P_LT\).
When the variable you want is in the denominator
Worked example 3 — Recover the total student count
Original formula — Main 9: \(M=\frac{M_T}{T}\)
\(M\) is the average serving time per student, in minutes per student. To solve for \(T\), multiply both sides by \(T\) to remove the denominator, then divide by \(M\):
Total serving time divided by minutes per student gives the number of students.
In Main 8, \(D_F=\frac{T\cdot P_L}{F}\) is the average lunch revenue per food server, in dollars per server. Multiply both sides by \(F\), then divide by \(T\), to obtain \(P_L=\frac{D_FF}{T}\).
Writing a rate from its units
Main 10(a) asks for average students served each minute. Put students in the numerator and minutes in the denominator: \(\frac{T}{M_T}\). Compare this with \(M=\frac{M_T}{T}\), which measures minutes per student. Reversing the fraction changes the meaning.
The question asks for “the average number of minutes students wait in the pizza line.” It is unclear whether this means time per student or the total time for the pizza students. The likely intention is to use the average serving time per student, \(M=\frac{M_T}{T}\), assuming similar serving times for the different lunch options.
Per pizza student: \(\frac{M_T}{T}=M\) minutes per student
Total for the pizza students: \(P\frac{M_T}{T}=PM\) minutes of serving time
“Serving time” would be clearer than “waiting time.” The question should also specify which of these two quantities is wanted.
2. Ready — Isolate a variable
Use the formula-solving method above to isolate the requested variable, then describe the operations you used. Try the problems before checking the key. The formulas below assume any quantity used as a divisor is nonzero.
Check your work — Ready 1–10 answer key
\(x=\frac7a\)
Divide both sides by \(a\).
\(p=w-8\)
Subtract 8 from both sides.
\(m=\frac{e}{c^2}\)
Divide both sides by \(c^2\).
\(t=\frac dr\)
Divide both sides by \(r\).
\(r=\frac dt\)
Divide both sides by \(t\).
\(h=7\)
Add \(h\) to both sides.
\(b=14\)
Add 11 to both sides.
\(y=2k\)
Multiply both sides by 2.
\(h=\frac{2A}{b}\)
Multiply both sides by 2, then divide both sides by \(b\).
\(x=\frac{y-b}{m}\)
Subtract \(b\) from both sides, then divide both sides by \(m\).
3. Set — Interpret quantities and units
Use Jaxon’s variable descriptions to explain each expression, including its units. For example, adding running distances gives a total distance; averaging three running times gives a time. Try Set independently, then check your explanations below.
Check your work — Set 11–17 answer key
Units column: \(M\), \(W\), \(F\), and \(S\) are measured in miles; \(H\) in hours; \(t_M\), \(t_W\), and \(t_F\) in minutes.
\(M+W+F\)
Total miles run in P.E. during one week.
\(4(M+W+F)\)
Total miles run in P.E. over four weeks with the same weekly distances.
\(2S\)
The round-trip distance between home and school, in miles.
\(t_M+t_W+t_F\)
Total minutes spent running in P.E. during one week.
\(\frac{t_M+t_W+t_F}{3}\)
Average running time per P.E. class, in minutes.
\(5(2H)\)
Total hours traveling to and from school during a five-day school week, using \(H\) hours for each one-way trip.
\(M+H\)
This adds miles to hours, so it does not describe a meaningful total.
4. Go — Set notation, intervals, and number lines
Describe a set using a rule
A set is a collection of values. You can describe a small set by listing its members, such as \(\{1,2,3\}\). But you cannot list every real number in an interval: there are infinitely many. Set-builder notation describes a set by giving a rule for which values belong to it.
| Part of the notation | Meaning |
|---|---|
| \(\{\ \}\) | Braces enclose the description of a set. |
| \(x\) | The first \(x\) names the values in the set. The conditions after the bar tell us which values to include. |
| \(\mid\) | The vertical bar means “such that.” |
| \(x\in\mathbb R\) | \(\in\) means “is a member of,” and \(\mathbb R\) is the set of real numbers—all numbers on the number line, including negative numbers, fractions, and decimals. |
| \(0\le x<5\) | Include values greater than or equal to 0 and less than 5. The comma before this condition means that it must also be satisfied. |
Worked example — Which values belong?
For \(\{x\mid x\in\mathbb R,\ 0\le x<5\}\):
- 0 belongs because \(0\le x\) includes 0.
- 2 and 1.5 both belong: each is greater than or equal to 0 and less than 5.
- 5 does not belong because \(x<5\) excludes 5.
- −1 does not belong because it is less than 0.
This is the same set as the interval \([0,5)\): the square bracket includes 0, and the parenthesis excludes 5. Both notations describe all the real numbers between the endpoints, not just whole numbers.
Move between set-builder and interval notation
To convert to interval notation, identify the lower and upper bounds and whether each is included. To convert back, write those same bounds as an inequality inside braces, with \(x\in\mathbb R\). For example:
Both endpoints are included, so both notations use inclusive symbols. For more on interval notation, see Section 3.2.
On a number line, use a filled circle for an included endpoint and an open circle for an excluded one. Shade the values that satisfy the inequality. Try the exercises before opening the key.
Check your work — Go 18–31 answer key
Convert between notations — 18–24
- \((-2,6)\)
- \(\{x\mid x\in\mathbb R,\ -4\le x\le7\}\)
- \([-9,\infty)\)
- \(\{x\mid x\in\mathbb R,\ 0<x\le13\}\)
- \([-15,-8]\)
- \(\{x\mid x\in\mathbb R,\ -32\le x<-15\}\)
- \(\{x\mid x\in\mathbb R\}\) (all real numbers)
Whole-number domains — 25
25. Set notation. For example, \(\{0,1,2,3\}\) lists only whole numbers; the interval \([0,3]\) also includes numbers such as \(1.5\).
Number-line graphs — 26–31
Plots not shown. The endpoint and shading descriptions below let you check your graphs.
- Open circle at 6; shade to the left
- Open circle at 5; shade to the right
- Filled circle at −9; shade to the right
- Filled circle at −7 and open circle at 0; shade between them
- Filled circles at 3 and 25; shade between them
- Open circle at −15 and filled circle at 8; shade between them
Additional learning resources
| Resource | What it supports |
|---|---|
| Khan Academy — Rearrange formulas to isolate specific variables | Using inverse operations to solve a formula for a chosen variable; supports the main task and Ready. |
| Khan Academy — Intro to dimensional analysis | Using units to make sense of distance, rate, and time. Includes unit conversion as well as the unit reasoning used here. |
Teacher’s review of the source packet
Main 10(b) · p. 7: The wording does not clearly distinguish time per pizza student from total time for the pizza students, and uses “waiting” where “serving” appears to be intended. The clarification gives both interpretations, assuming similar serving times across lunch options.
Teacher guidance on Main 10: The notes describe Question 10 as an optional extension and invite students to propose expressions, but do not give an interpretation or solution for 10(b). They mention a separate answer-key page that is not included in this local teacher edition.
Teacher notes: The purpose paragraph reverses the division in its gasoline-efficiency example: miles per gallon requires miles divided by gallons, not gallons divided by miles. The student-facing explanation here uses the correct order for rates.
Coverage — Section 4.2
| Packet location | Where supported |
|---|---|
| Main 1–7 · pp. 5–6 | Variable meanings and units; worked solutions for salad count and lunch price; total student count |
| Main 8–10 · p. 7 | Revenue per server, solving for a denominator variable, rate meanings, and the waiting-time clarification |
| Ready 1–10 · p. 8 | Formula-solving instruction and answer key with operation descriptions |
| Set 11–17 · p. 8 | Unit reasoning and an explanation for each answer |
| Go 18–31 · p. 9 | Set-builder notation explained symbol by symbol, a membership example, interval conversion, and an answer key with graph descriptions instead of plots |
Source: Mathematics Vision Project, Secondary Mathematics One, Module 4: Equations and Inequalities (May 2016), Section 4.2, printed pp. 5–9; teacher notes for 4.2. Packet equations and short excerpts are attributed under CC BY 4.0. Equations use MathJax.
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