Chapter 1. Solving Algebraic Equations for a Variable

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1 CHAPTER 1 Solving Algebraic Equations for a Variable Here you ll learn how to isolate the variable in an equation or formula. Problem: You are planning a trip to Spain in the summer. In the U.S., we use Fahrenheit to measure the temperature, but in the temperature in Celsius for C and solve the equation for F. When you start looking at the weather in Spain in the summer, the website says it is usually C. What should you pack? Guidance Solving an algebraic equation for a variable can be tricky, but it can also be very useful. This technique can be used to go back and forth between two different units of measurement. To solve for, to isolate, a variable within an equation, you must undo the operations that are in the equation. Example A Solve for x in the equation x y = 1. Solution: To solve for x, you need to move the y to the other side of the equation. In order to do this, we must do the opposite operation. Since the y is being subtracted, we must add it to the other side of the equation. x y = 1 +y = +y x = y + 1 Now we need to get x alone. x is the same as multiplied by x. Therefore, to undo the multiplication we must divide both sides of the equation by. x y 1 = + x = y + When undoing multiplication, you must divide everything in the equation by. A few things to note: 1. To undo an operation within an equation, do the opposite.. Perform the opposite operations in the reverse order of the Order of Operations.. Combine all like terms on the same side of the equals sign before doing #1. 1

2 Example B Given the equation b + 6a = 8, find b when a = 1 and a =. Solution: This example combines what was learned in the last section with what we did in the previous example. First, isolate b. Move both the 6a and the over to the other side by doing the opposite operation. b + 6a = 8 6a + = + 6a b = 1 6a Now, we have a fraction multiplied by b. To undo this we must multiply by the reciprocal of / which is /. This means, we must multiply everything in the equation by the reciprocal. b = 1-6a b = 1 6a b = 18 a Now, we can do what the question asks, find b when a = 1 and -. Plug in these values for a. a = 1 : b = 18 (1) = 18 = a = : b = 18 ( ) = = 6 Example C The area of a triangle is A = 1 bh, where b is the base of the triangle and h is the height. You know that the area of a triangle is 60 in and the base is 1 in. Find the height. Solution: First solve the equation for h. 1 A = bh ()(A) = 1bh Multiply both sides by A = h Divide both side by b. b Now, plug in what we know to find h. (60) = 10 = 10. The height is 10 inches. 1 1 This process is helpful when solving for any variable in any equation or formula. Some formulas you may need to know for this text are listed below.

3 Distance Temperature d = rt F = C + TABLE 1.1: d = distance, r = rate, t = time F = degrees in Fahrenheit, C = degrees in Celsius A = area, b = base, h = height Area of a Triangle A = 1 bh Area of a Rectangle A = bh A = area, b = base, h = height Area of a Circle A = πr A = area, r = radius Area of a Trapezoid A = 1 h(b1 + b) A = area, h = height, b1 = one base, b = other base Perimeter of a Rectangle P = l + w P = perimeter, l = length, w = width Circumference of a Circle C = πr C = circumference, r = radius Intro Problem Revisit First, let s substitute in C = to find the degrees in Fahrenheit. F = + F = 6 + F = As you can see, a good estimator for the conversion is to double the degrees, when in Celsius, and then add. Needless to say, it is going to be quite hot in Spain and you should pack summer clothes. Guided Practice 1. Solve x y = 1 for x.. If the temperature is 1 F, what is it in Celsius? Answers 1. First add y to both sides, then multiply both sides by the reciprocal of / /x y = 1 + y = 1 + y /x = 1 + y /(/ x) = (1 + y) x = 0 + 8/y

4 . You can solve this problem two different ways: First, plug in 1 for F and then solve for C or solve for C first and then plug in 1 for F. We will do the second option. F = C + F = /C / (F ) = C Now, plug in 1 for F. (1 ) C = C = C =. Homework Solve the following equations or formulas for the indicated variable. 1. 6x y = ; solve for y.. c + d = 16; solve for c.. f 6g = 1; solve for f.. 1/x + y = 1, solve for x. m + n = ; solve for n. 6. / m + /n =, solve for m 7. P = l + w; solve for w. 8. F = /C + ; solve for C. Find the value of y given the value of x.. x 8y = ; x = y x = 1; x = x + 1/y = -; x = /x + /y -18 = 0; x=-

5 For questions 1-1, use the formulas in the chart above to answer the following questions. 1. If the temperature is 86 F, what is it in Celsius? 1. If the area of a circle is 6π cm, what is the radius? 1. The area of a trapezoid is 7 f t, the height is 8 ft and b1 is 6, what is the length of the other base? For questions 16-17, use the equation for the surface area of a cylinder, SA = πr + πrh, where r is the radius and h is the height. 16. Solve the equation for h. 17. Find h if the surface area is 10π cm and the radius is 6 cm. 18. Challenge The formula for the volume of a sphere is V = πr. Solve the equation for r, the radius.