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Algebra 1 9-12 / Week 02 / Friday
5/6
Week 02 · Equations and Inequalities in One Variable

Friday

Rearrange the formula
// Keep both sides equal, and know why
⏱ about 20 min

Friday: Rearrange the Formula

Wren writes d = rt on the whiteboard. "Distance equals rate times time. Our cart rule from last week."

"We know the distance and the speed," Comet says. "We want the time. So solve for t."

"Same moves as beans," Wren says. "Divide both sides by r. t = d ÷ r. What do you notice?"

"The letters moved but nothing new happened," Comet says. "Equal things divided by r stay equal."

Nova projects the frame formula, p = 2l + 2w. "Would you like a hint? Which letter do you want alone?"

"The width," Comet says. "Subtract 2l from both sides. p - 2l = 2w. Divide by 2. w = (p - 2l) ÷ 2."

"Check with numbers," Wren says. "150 minus 90 is 60, over 2 is 30. The frame width. It works."

A formula is an equation with several letters

d = rt holds for every steady run: distance, rate and time. Solving it for t means getting t alone.

Divide both sides by r: t = d ÷ r. Check with Loft numbers: 240 ÷ 150 = 1.6 seconds.

The moves are the ones from Tuesday. Each is allowed because equal quantities treated the same stay equal.

FormulaSolve forStepsResult
d = rttDivide both sides by r.t = d ÷ r
p = 2l + 2wwSubtract 2l from both sides, then divide by 2.w = (p - 2l) ÷ 2
v = 3t + 15tSubtract 15 from both sides, then divide by 3.t = (v - 15) ÷ 3
ax + b = cxSubtract b from both sides, then divide by a.x = (c - b) ÷ a

Letters as coefficients

In ax + b = c, the letters a, b and c stand for numbers you will fill in later. Solve for x exactly as if they were numbers.

Subtract b, then divide by a: x = (c - b) ÷ a. With a = 4, b = 6 and c = 38, that is x = 8, the screw count again.

The only care needed: a cannot be 0, because you cannot divide by 0.

Equations and limits in everyday life

  • A phone timer counting down from 30 minutes: 30 - m is the time left after m minutes.
  • A shelf rated for a limit: box plus parts must stay at or under it. That is an inequality.
  • Any steady trip: d = rt, solved for whichever of the three you do not know.

The week in five lines

  • Name the unknown, write what the story says is true, then solve.
  • Every solving step is allowed because equal things treated the same stay equal.
  • Check a solution by substitution, in the equation and in the story.
  • An inequality gives a range. Flip the sign only when you multiply or divide by a negative.
  • A formula can be solved for any one letter with the same moves.
REARRANGE THE FORMULA
  • Read the question.
  • Tap your answer.
The cart rule is d = rt. Solve it for the time, t. Which form is right?
The frame rule is p = 2l + 2w. Solve it for the width, w. Which form is right?
The jug rule is v = 3t + 15. Solve it for the time, t. Which form is right?
Solve ax + b = c for x, treating a, b and c as numbers. Which form is right?
MIXED REVIEW
  • Read the question.
  • Tap your answer.
Use t = d ÷ r with d = 90 centimeters and r = 30 centimeters per second. How many seconds is t?
Use w = (p - 2l) ÷ 2 with p = 150 and l = 45 centimeters. What is w?
What is x if 5x - 7 = 18?
A tower rule: 4c + 9 ≤ 33 cups. Which values of c fit?
StatementTrue or false?
Solving d = rt for t gives t = d ÷ r.?
In ax + b = c, the letter a may be any number, including 0.?
x = 5 is a solution of 5x - 7 = 18.?
(38 - 6) ÷ 4 = 8?
Multiplying both sides of an inequality by a positive number flips the sign.?
WHY THIS EXERCISEThese five lines are the whole week. Next week the crank box turns inputs into outputs, and functions begin.
Try it
Pick a formula you know, such as the distance around a rectangle. Solve it for a different letter.
Check your new form with real measurements from something at home.
Draw d = rt three times, each solved for a different letter, with arrows showing the move you made.

Terrific week. You can write, solve, justify and rearrange. Tomorrow is Loft Day.

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