Learning Target: Scholars Will Solve Multi-Variable Equations for One of the Variable

Lesson 3.3.2

HW: 3-93 to 3-98

Learning Target: Scholars will solve multi-variable equations for one of the variable.

So far in this course you have solved several types of equations with one variable. Today you will apply your equation-solving skills to rewrite equations with two or more variables. 3-87. You now have a lot of experience working with equations that compare two quantities. For example, while working on the Big Race, you found relationships of the form y = mx + b which compared x (time in seconds) with y (distance in meters). If a participant's race can be modeled with the equation y = 3x + 4: 1. How much of a head start did the participant get? How can you tell from the equation? 2. What was the participant's rate of speed? That is, how fast did she go? Justify your answer.

3-88. CHANGING FORMS You could find the slope and starting value for y = 3x + 4 quickly because the equation is in y = mx + b form. But what if the equation is in a different form? Explore this situation below. 3. The line −6x + 2y = 10 is written in standard form. Can you easily tell what the slope of the line is? Its starting value? Predict these values.

4. The equation −6x + 2y = 10 is shown on the Equation Mat at right. Set up this equation on your Equation Mat using tiles. Using only “legal” moves, rearrange the tiles to get y by itself on the left side of the mat. Record each of your moves algebraically. 5. Now use your result from part (b) to find the slope and starting value of the line −6x + 2y = 10 . Did your result match your prediction in part (a)?

3-89. Many times in real-world situations a formula with more than one variable may not be in the form you need. The previous problem showed that standard form linear equations do not show the slope and y- intercept until they are solved for y, that is, until y is isolated on one side of the equation. The formulas in this problem are used in many different jobs. Sometime you need to solve them for a different variable in order for the formula to be useful. Solve each formula for the given variable. 6. W = Fd. Find the force, F, needed to move a piano given the amount of work applied, W, and distance moved, d.

7. . Find the temperature in Celsius, C, when given the temperature in degrees Fahrenheit, F.

8. . The symbol, , is a letter of the Greek alphabet. Sometimes scientists use

Greek letters for variables. Find the mass, m, of a precious stone given its density, , and volume, V.

9. . Find, r, the distance to the light bulb, given I, the intensity of light, and W, the wattage of the light bulb.

3-90. Solve each of the following equations for the indicated variable. Use your Equation Mat if it is helpful. Write down each of your steps algebraically. 10. Solve for y: 2(y − 3) = 4 11. Solve for x: 2x − 6y = 12 12. Solve for y: 6x + y = 2y + 8 13. Solve for x: 3(2x + 4) = 2 + 6x + 10

3-93. Solve each equation. Be sure to find all possible answers and check your solutions.

3-94. Solve each equation below for the indicated variable. 5. 3x − 2y = 18 for x 6. 3x − 2y = 18 for y 7. rt = d for r

8. for r

3-95. Evaluate the following rational expressions.

3-96. Find the equation of each line described below. 3-96 HW eTool (Desmos). A line with slope of 0 that passes through the point (6, –11). 13. A line that passes through the points (12, 12) and (20, 6).

3-97. Graph the lines y = −4x + 3 and y = x − 7 on the same set of axes. Then find their point of intersection. 3-97 HW eTool (Desmos).

3-98. Simplify each expression using the laws of exponents. 14. (x2)(x2y3)

15. 16. (2x2)(−3x4) 17. (2x)3

3-99. One way to solve absolute value equations is to think about “looking inside” the absolute value. The “inside” must be positive or negative, so you should solve the equation both ways. For example, you could record your steps as shown at right. Solve each equation. Be sure to find all possible answers and check your solutions.

4. 3-100. Find each of the following products by drawing and labeling a generic rectangle or by using the Distributive Property. 1. 5x(x − 6) 2. −9y(6 − 3y)

3-101. For each generic rectangle below, find the dimensions (length and width). Then write the area as a product of the dimensions and as a sum.

3-102. Solve each of the following equations. Be sure to show your work carefully and check your answers. 1. 2(3x − 4) = 22 2. 6(2x − 5) = −(x + 4) 3. 2 − (y + 2) = 3y 4. 3 + 4(x + 1) = 159

3-103. Multiply each of the following expressions. Show all of your work. 1. (x + 3)(4x + 5) 2. (−2x2 − 4x)(3x + 4) 3. (3y − 8)(−x + y) 4. (y − 4)(3x + 5y − 2)

3-104. Solve each of the following equations for the indicated variable. Show all of your steps. 1. y = 2x − 5 for x 2. p = −3w + 9 for w 3. 2m − 6 = 4n + 4 for m 4. 3x − y = −2y for y

Lesson 3.3.2  3-87. See below: 1. 4 meters; the y-intercept 4, or when x = 0, y = 4. 2. 3 meters per second; the rate is the slope which is the coefficient of the x.  3-88. See below: 1. Depending on their experience in previous courses, students at this point will probably predict that the slope is –6 and the y-intercept is (0, 10). 2. y = 3x + 5 3. The y-intercept is (0, 5); the slope is 3.  3-89. See below: 1. 2. 3. 4.  3-90. See below: 1. y = 5 2. x = 3y + 6 3. y = 6x − 8 4. x = any number  1.   3-93. See below: 1. x = ±7 2. x = ±16 3. x = 3, −17 4. x = ±52.1  3-94. See below: 1. x = 6 + y 2. y = x − 9 3. r = 4. r =  3-95. See below: 1. 5 2. 3. 4.  3-96. See below: 1. y = −11 2. y = −x + 21  3-97. See graph below. (2, –5)   3-98. See below: 1. x4y3 2. xy 3. –6x6 4. 8x3  3-99. See below: 1. x = 10 or x = –16 2. x = , – 3. x = – or x = 6 4. no solution  3-100. See below: 1. 5x2 – 30x 2. –54y + 27y2  3-101. See below: 1. 2x(x + 5) = 2x2 + 10x 2. (2x + 5)(x + 3) = 2x2 + 13x + 15  3-102. See below: 1. x = 5 2. x = 2 3. x = 0 4. x = 38  3-103. See below: 1. 4x2 + 17x + 15 2. –6x2 – 20x – 16 3. –3xy + 3y2 + 8x – 8y 4. 3xy + 5y2 – 22y – 12x + 8y  3-104. See below: 1. 2. w = 3. m = 4. y = –3x