In This Chapter
Getting down to the basics when solving linear equations
Making grouping symbols and fractions work
Putting proportions in their place
Linear equations are algebraic equations that have no powers on the variables that are greater than the first power. The most common (and easiest) way of solving linear equations is to perform operations or other manipulations so that the variable you’re solving for is on one side of the equation and the numbers or other letters and symbols are on the other side of the equation. You want to get the variable alone so that you can finish with a statement, such as x = 4 or y = 2a. Because linear equations involve just the first degree (power one) of the variable, you look for just one answer.
This chapter provides you with the different setups or situations where linear equations are usually found — and what to do when you find them.
Using the Addition/Subtraction Property
One of the most basic properties of equations is that you can add or subtract the same amount from each side of the equation and not change the balance or equality. The equation is still a true statement (as long as it started out that way) after adding or subtracting the same from each side. You use this property to get all the terms with the variable you want to solve for to one side and all the other letters and numbers to the other side so that you can solve the equation for the value of the variable.
You can check the solution by putting the answer back in the original equation to see whether it gives you a true statement.
Q. Solve for x: x + 7 = 11
A. x = 4. Subtract 7 from each side (same as adding –7 to each side), like this:
Q. Solve for y: 8y – 2 = 7y – 10
A. y = –8. First add –7y to each side to get rid of the variable on the right (this moves the variable to the left with the other variable term) and then add 2 to each side to get rid of the –2 (gets the numbers together on the right). This is what the process looks like:
1. Solve for x: x + 4 = 15.
Solve It
2. Solve for y: y – 2 = 11.
Solve It
3. Solve for x: 5x + 3 = 4x – 1.
Solve It
4. Solve for y: 2y + 9 + 6y – 8 = 4y + 5 + 3y – 11.
Solve It
Using the Multiplication/Division Property
Another property of equations is that when you multiply or divide both sides by the same number (but not zero), then the equation is still an equality; it’s still a true statement. You can use this multiplication/division property alone or with the addition/subtraction property to help solve equations for the value of the variable.
Q. Solve for x: –3x = –45.
A. x = 15. Divide each side by –3 to determine what x is:
Q. Solve for y: .
A. y = 60. Multiply each side by 5 to solve the equation for y:
.
5. Solve for x: 6x = 24.
Solve It
6. Solve for y: –4y = 20.
Solve It
7. Solve for z: .
Solve It
8. Solve for w: .
Solve It
Putting Several Operations Together
The different properties of equations that allow you to add the same number to each side or multiply each side by the same number (except zero) are the backbone of solving linear equations. More often than not, you have to perform several different operations to solve a particular equation. In the cases where the problem just has the operations of addition, subtraction, multiplication, and division (and there aren’t any grouping symbols to change the rules), you first do all the addition and subtraction to get the variables on one side and the numbers on the other side. Then you can multiply or divide to get the variable by itself.
The side you move the variable to really doesn’t matter. Many people like to have the variable on the left, so you can read x = 2 as “x equals 2.” Writing 2 = x is just as correct. You may prefer having the variable on one side in one equation and on the other side in another equation — depending on which side makes for less awkward operations or keeps the variable with a positive factor.
Q. Solve for x: 6x – 3 + 2x = 9x + 1 – 4x + 8.
A. x = 4
1. Simplify each side of the equation by combining like terms.
8x – 3 = 5x + 9
2. Add 3 to and subtract 5x from each side.
3. Divide each side by 3 to get x = 4.
Q. Solve for y: .
A. –6 = y
1. By subtracting from each side and subtracting 5 from each side, you get
2. Multiply each side by 3 and then divide each side by 2.
Another way to do the last two operations in just one is to multiply each side of the equation by the reciprocal of , which is .
9. Solve for x: 3x – 4 = 5.
Solve It
10. Solve for y: .
Solve It
11. Solve for x: 5x – 3 = 8x + 9.
Solve It
12. Solve for z: .
Solve It
13. Solve for y: 4y + 16 – 3y = 7 + 3y.
Solve It
14. Solve for x: .
Solve It
Solving Linear Equations with Grouping Symbols
The most general procedure to use when solving linear equations is to add and subtract first and then multiply or divide. This general rule is interrupted when the problem contains grouping symbols such as ( ), [ ], { }. (See Chapter 2 for more on grouping symbols.) If the equation has grouping symbols (and this includes fractions with many-termed numerators), you need to perform whatever operation is indicated by the grouping symbol before carrying on with the other rules.
If you perform an operation on the grouping symbol, then every term in the grouping symbol has to have that operation performed on it.
Q. Solve 8(2x + 1) + 6 = 5(x – 3) + 7.
A. x = –2
1. Distribute the 8 over the two terms in the left parentheses and the 5 over the two terms in the right parentheses.
You get the equation 16x + 8 + 6 = 5x – 15 + 7.
2. Combine the two numbers on each side of the equation.
You get 16x + 14 = 5x – 8.
3. Subtract 5x and 14 from each side; then divide each side by 11.
Q. Solve .
A. x = –3 First multiply each term on both sides of the equation by 4:
Combine the like terms on the left. Then subtract 4x from each side and subtract 7 from each side; finally, divide each side by –3.
15. Solve for x: 3(x – 5) = 12.
Solve It
16. Solve for y: 4(y + 3) + 7 = 3.
Solve It
17. Solve for x: .
Solve It
18. Solve for y: 5(y – 3) – 3(y + 4) = 1 – 6(y – 4).
Solve It
19. Simplify the quadratic equation to create a linear equation and then solve for x: x(3x + 1) – 2 = 3x2 – 5.
Solve It
20. Simplify the quadratic equation to create a linear equation and then solve for x: (x – 3)(x + 4) = (x + 1)(x – 2).
Solve It
Working It Out with Fractions
Fractions aren’t everyone’s favorite thing, although it’s hard to avoid them in everyday life or in algebra. Fractions in algebraic equations can complicate everything, so just getting rid of fractions is often easier than trying to deal with finding common denominators several times in the same problem.
Two general procedures work best when dealing with algebraic fractions.
If you can easily isolate a single term with the fraction on one side, do the necessary addition and subtraction, and then multiply each side of the equation by the denominator of the fraction (be sure to multiply each term in the equation by that denominator).
If the equation has more than one fraction, find a common denominator for all the terms and then multiply each side of the equation by this common denominator. Doing so, in effect, gets rid of all the fractions.
&
nbsp; Q. Solve by isolating the fractional term on the left.
A. x = 5
1. Subtract 4 from each side.
2. Multiply each side by 3.
Now the problem is in a form ready to solve.
3. Add 1 to each side and divide each side by 2.
Q. This time, use the second procedure from the bulleted list to solve .
A. x = –8
1. Determine the common denominator for the fractions, which is 4.
2. Multiply each term in the equation by 4.
Doing so eliminates all the fractions after you reduce them.
3. Add 6x to each side and subtract 60 from each side. Then finish solving by dividing by 7.
21. Solve for x: .
Solve It
22. Solve for x: .
Solve It
23. Solve for y: .
Solve It
24. Solve for x: .
Solve It
25. Simplify the rational equation by multiplying each term by y. Then solve the resulting linear equation for y: .
Solve It
26. Simplify the rational equation by multiplying each term by z. Then solve the resulting linear equation for z: .
Solve It
Solving Proportions
A proportion is actually an equation with two fractions set equal to one another. The proportion has the following properties:
The cross products are equal: ad = bc.
If the proportion is true, then the flip of the proportion is also true: .
You can reduce vertically or horizontally: or .
You solve proportions that are algebraic equations by cross-multiplying, flipping, reducing, or using a combination of two or more of the processes. The flipping part of solving proportions usually occurs when you have the variable in the denominator and can do a quick solution by first flipping the proportion and then multiplying by a number.
Q. Solve for x: .
A. x = 10
1. First, reduce horizontally through the denominators. Then cross-multiply.
2. Subtract 12x from each side. Then divide each side by 2.
Q. Solve for x: .
A. x = 25
1. Flip the proportion to get .
2. Reduce through the denominators and then multiply each side by 2. (You could reduce vertically in the first fraction, too, but dividing by 6 in the denominators is a better choice.)
27. Solve for x: .
Solve It
28. Solve for y: .
Solve It
29. Solve for z: .
Solve It
30. Solve for y: .
Solve It
Answers to Problems on Solving Linear Equations
This section provides the answers (in bold) to the practice problems in this chapter.
1. Solve for x: . The answer is x = 11.
2. Solve for y: . The answer is y = 13.
3. Solve for x: . The answer is .
4. Solve for y: . The answer is . By combining like terms, you get
5. Solve for x: . The answer is .
6. Solve for y: . The answer is .
7. Solve for z: . The answer is .
8. Solve for w: . The answer is .
9. Solve for x: . The answer is .
10. Solve for y: . The answer is .
11. Solve for x: . The answer is .
12. Solve for z: . The answer is .
Note that .
13. Solve for y: . The answer is . Combine like terms and then solve:
14. Solve for x: .
15. Solve for x: . The answer is .
16. Solve for y: . The answer is .
17. Solve for x: . The answer is .
18. Solve for y: . The answer is .
19. Solve for x: . The answer is .
20. Solve for x: . The answer is .
21. Solve for x: . The answer is .
22. Solve for x: . The answer is . Twelve is a common denominator, so
23. Solve for y: . The answer is . Twenty is a common denominator, so
24. Solve for x: . The answer is . A common denominator is 6, so
25. Solve for y: . The answer is . The common denominator is y, so
26. Solve for z: . The answer is . A common denominator is 6z, so
27. Solve for x: . The answer is . Reduce by dividing the denominators by 4 and the right fractions by 3. Then cross-multiply.
28. Solve for y: . The answer is . Reduce through the numerators. Then flip.
29. Solve for z: . The answer is z = 16. Reduce the fraction on the right by dividing the numerator and denominator by 7. Then reduce through the denominators before you cross-multiply:
30. Solve for y: . The answer is . Flip to get and solve by reducing the fraction on the left and then cross-multiplying:
Yes, you could also have reduced through the denominators, but when the numbers are small enough, it’s just as quick to skip that step.
Chapter 13
Muscling Up to Quadratic Equations
In This Chapter
Taking advantage of the square root rule
Solving quadratic equations by factoring
Enlisting the quadratic formula
Completing the square to solve quadratics
Dealing with the impossible
A quadratic equation is an equation that is usually written as ax2 + bx + c = 0 where b, c, or both b and c may be equal to 0, but a is never equal to 0. The solutions of quadratic equations can be two real numbers, one real number, or no real number at all. (Real numbers are all the whole numbers, fractions, negatives and positives, radicals, and irrational decimals. Imaginary numbers are something else again!)
When solving quadratic equations, the most useful form for the equation is one in which the equation is equal to 0 and the terms are written in decreasing powers of the variable. When set equal to zero, you can factor for a solution or use the quadratic formula. An exception to this rule, though, is when you have just a squared term and a number, and you want to use the square root rule. (Say that three times quickly.) I go into further detail about all these procedures in this chapter.
For now, strap on your boots and get ready to answer quadratic equations. This chapter offers you plenty of chances to get your feet wet.
Using the Square Root Rule
You can use the square root rule when a quadratic equation has just the squared term and a number — no term with the variable to the first degree. This rule says that if x2 = k, then , as long as k isn’t a negative number. The tricky part here — what most people trip on — is in remembering to use the “plus or minus” symbol so that both solutions are identified.
Algebra I Workbook For Dummies Page 11