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**Warm Up Solve each equation. 1. 5 + x = –2 2. 8m = 43 3. –7**

5. 10k – 6 = 9k + 2 –7 19 3 8

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**Objectives Solve a formula for a given variable.**

Solve an equation in two or more variables for one of the variables.

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Vocabulary formula literal equation

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**A formula is an equation that states a rule for a relationship among quantities.**

In the formula d = rt, d is isolated. You can "rearrange" a formula to isolate any variable by using inverse operations. This is called solving for a variable.

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Solving for a Variable Step 1 Locate the variable you are asked to solve for in the equation. Step 2 Identify the operations on this variable and the order in which they are applied. Step 3 Use inverse operations to undo operations and isolate the variable.

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Example 1 Solve the formula d = rt for t d = rt Locate t in the equation. Since t is multiplied by r, divide both sides by r to undo the multiplication.

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Example 2 The formula for the area of a triangle is A = bh, where b is the length of the base, and is the height. Solve for h. A = bh Locate h in the equation. Since bh is multiplied by , divide both sides by to undo the multiplication. 2A = bh Since h is multiplied by b, divide both sides by b to undo the multiplication.

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**Dividing by a fraction is the same as multiplying by the reciprocal.**

Remember!

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Example 3 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

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Example 3 Continued 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. Since e is multiplied by –10, divide both sides by –10 to undo the multiplication.

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**Dividing by a fraction is the same as multiplying by the reciprocal.**

Remember!

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Example 4 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

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**A formula is a type of literal equation**

A formula is a type of literal equation. A literal equation is an equation with two or more variables. To solve for one of the variables, use inverse operations.

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Example 5 A. Solve x + y = 15 for x. x + y = 15 Locate x in the equation. –y –y x = –y + 15 Since y is added to x, subtract y from both sides to undo the addition. B. Solve pq = x for q. pq = x Locate q in the equation. Since q is multiplied by p, divide both sides by p to undo the multiplication.

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Example 6 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.

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Example 7 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.

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Lesson Summary: 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

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Lesson Summary: 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

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