1) Solve 2x - 4y = 7 for x + 4y + 4y 2x = 7 + 4y 2 2 Draw “the river” D U 1) Solve 2x - 4y = 7 for x · 2 -4y +4y ÷ 2 + 4y + 4y 2x = 7 + 4y 2 2 Draw “the river” Add 4y to both sides Simplify Divide both sides by 2 Does it simplify? This fraction cannot be simplified because both terms in the numerator are not divisible by 2.

Let’s look at another example: 2Q-c = d Multiply both sides by 2.        Subtract ‘c’ from each side.

2) Solve 2x - 4y = 7 for y - 2x - 2x -4y = 7 - 2x -4 -4 D U 2) Solve 2x - 4y = 7 for y · -4 +2x -2x ÷ -4 - 2x - 2x -4y = 7 - 2x -4 -4 Draw “the river” Subtract 2x from both sides Simplify Divide both sides by -4 Does it simplify? Nope!

Example 2B: Solving Formulas for a Variable The formula for a person’s typing speed is ,where s is speed in words per minute, w is number of words typed, e is number of errors, and m is number of minutes typing. Solve for e. Locate e in the equation. Since w–10e is divided by m, multiply both sides by m to undo the division. ms = w – 10e Since w is added to –10e, subtract w from both sides to undo the addition. –w –w ms – w = –10e

3) Solve for y: y + a = 3c -a -a y = 3c - a Draw “the river” D U 3) Solve for y: + a ÷ 3 · 3 - a Draw “the river” Clear the fraction – multiply both sides by 3 Simplify Subtract a from both sides y + a = 3c -a -a y = 3c - a

Check It Out! Example 2 The formula for an object’s final velocity is f = i – gt, where i is the object’s initial velocity, g is acceleration due to gravity, and t is time. Solve for i. f = i – gt Locate i in the equation. f = i – gt + gt +gt Since gt is subtracted from i, add gt to both sides to undo the subtraction. f + gt = i

4) Solve for y: 4x – 2y = 12 - 4x - 4x -2y = 12 – 4x -2 -2 y = -6 + 2x D U 4) Solve for y: 4x – 2y = 12 · -2 + 4x - 4x ÷ -2 - 4x - 4x -2y = 12 – 4x -2 -2 y = -6 + 2x or y = 2x - 6 Draw “the river” Subtract 4x from both sides Simplify Divide both sides by -2 Does it simplify? Yes!

Check It Out! Example 3a Solve 5 – b = 2t for t. 5 – b = 2t Locate t in the equation. Since t is multiplied by 2, divide both sides by 2 to undo the multiplication.

Check It Out! Example 3b Solve for V Locate V in the equation. Since m is divided by V, multiply both sides by V to undo the division. VD = m Since V is multiplied by D, divide both sides by D to undo the multiplication.

Lesson Quiz: Part 1 Solve for the indicated variable. 1. 2. 3. 2x + 7y = 14 for y 4. for h P = R – C for C C = R – P for m m = x(k – 6 ) 5. for C C = Rt + S

Lesson Quiz: Part 2 Euler’s formula, V – E + F = 2, relates the number of vertices V, the number of edges E, and the number of faces F of a polyhedron. 6. Solve Euler’s formula for F. F = 2 – V + E 7. How many faces does a polyhedron with 8 vertices and 12 edges have? 6