Checkpoint Sample

Sample Checkpoint

Checkpoint 6A
Problem 6-29

Rewriting Equations With More Than One Variable

Answers to problem 6-29: a: $y = \frac{3 x - 10}{5} = \frac{3}{5} x - 2$ , b: $x = \frac{y - b}{m}$ , c: $r^{2} = \frac{A}{π}$

Rewriting equations with more than one variable may be done in a variety of ways but normally you follow the same steps as you would for solving an equation with a single variable. Commonly, the first step is to multiply all the terms by a common denominator to remove all of the fractions. Then solve in the usual way. Collect like terms. Isolate the specified variable terms on one side and everything else on the other. Finally, divide or undo the exponent so that the variable is alone. The final answer will be an equation that involves variables and possibly numbers.

Example 1: Solve for y: $2 x + 3 y - 9 = 0$

Solution: 2x + 3y − 9 = 0

problem

$3 y = - 2 x + 9$

subtract $2 x$ , add $9$ on each side

$y = \frac{- 2 x + 9}{3}$

3 divide by 3

$y = - \frac{2}{3} x + 3$

simplify

Example 2: Solve for r: V = $\frac{4}{3}$ πr³

Solution: V = $\frac{4}{3}$ πr³

problem

3V = 4πr³

multiply by 3 on each side

$\frac{3 V}{4 π} = r 3$

divide by $4 π$

$\sqrt[3]{\frac{3 V}{4 π}} = r$

cube root

Now we can go back and solve the original problems.

$- 3 x + 5 y = - 10 5 y = 3 x - 10 y = \frac{3 x - 10}{5} y = x - 2$

$y = m x + b y - b = m x + b \frac{y - b}{m} = x$

$A = π r^{2} \frac{A}{π} = r^{2}$

Here are some more to try. Solve each equation for the specified variable.

$2 x - 3 y = 9 (for x)$

$2 x - 3 y = 9 (for y)$

$5 x + 3 y = 15 (for y)$

$4 n = 3 m - 1 (for m)$

$2 w + 2 l = P (for w)$

$2 a + b = c (for a)$

$I = \frac{E}{R} (for R)$

$y = \frac{1}{4} x + 1 (for x)$

$c 2 = a 2 + b 2 (for b 2)$

$V = s 3 (for s)$

$S = 4 π r 2 (for r 2)$

$m = \frac{a + b + c}{3} (for a)$

$a 3 + b = c 2 (for a)$

$\frac{F}{a} = m (for a)$

$A = \frac{1}{2} h (b 1 + b 2) (for b 1)$

$V = \frac{1}{3} π r 2 h (for h)$

Answers:

$x = \frac{3 y + 9}{2}$

$y = \frac{2 x - 9}{3}$

$y = \frac{15 - 5 x}{3}$

$m = \frac{4 n + 1}{3}$

$w = \frac{P - 2 l}{2}$

$a = \frac{c - b}{2}$

$R = \frac{E}{I}$

$x = 4 (y - 1)$

$b^{2} = c^{2} - a^{2}$

$s = \sqrt[3]{V}$

$r^{2} = \frac{S}{4 π}$

$a = 3 m - b - c$

$a = \sqrt[3]{c^{2} - b}$

$a = \frac{F}{m}$

$b_{1} = \frac{2 A}{h} - b_{2}$

$h = \frac{3 V}{π r^{2}}$

Sample Checkpoint

Checkpoint 6AProblem 6-29

Rewriting Equations With More Than One Variable

Checkpoint 6A
Problem 6-29