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By Root 294 0

from one language to another. Phrases are translated word by word at first, with care to observe syntax. Variables and numbers are nouns, operation signs(+, −, ×, ÷) act as conjunctions, and equal signs or inequality signs are verbs. Sometimes you need to learn an idiom. For example, “3 less than a number” doesn’t involve a less-than inequality sign. It translates to x − 3.

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EXERCISE 1.5

Simplify each expression by combining like terms where possible.

1. 3t + 8t

2. 10x − 6x

3. 5x + 3y − 2x

4. 2y − 3 + 5x + 8y − 4x

5. 6 − 3x + x2 − 7 + 5x − 3x2

6.(5t + 3)+(t − 12r)−8 + 9r +(7t − 5)

7.(5x2 − 9x + 7)+(2x2 + 3x + 12)

8.(2x − 7)−(y + 2x)−(3 + 5y)+(8x − 9)

9.(3x2 + 5x − 3)−(x2 + 3x − 4)

10. 2y −(3 + 5x)+ 8y −(4x − 3)

Write a variable expression for each phrase. Use the variable shown in parentheses at the end of the phrase.

11. Two more than 3 times a number(x)

12. Three times a number decreased by 7(y)

13. The quotient of a number and 3, increased by 11(t)

14. Eight less than the product of a number and 9(n)

15. The sum of a number and its opposite(w)

16. Three less than 5 times a number, divided by the square of the number(p)

17. The square of a number reduced by 4 times the number(r)

18. Eight more than the quotient of a number and 1 less than twice the number(x)

19. The product of 2 more than 3 times a number and 6 less than 4 times the number(z)

20. The square root of 4 times the square of a number decreased by 1(v)

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Evaluating expressions

Evaluate, if you take the word back to its roots, means to bring out or lead out the value. You have evaluated, or found the value of, an algebraic expression when you know what numbers the variables stand for. You solve an equation or inequality to find out what the value of the variable is. To evaluate an algebraic expression, replace each variable with its value and simplify according to the order of operations.

To evaluate when x = 2 and y= 1, replace each variable with its value: . The order of operations calls for parentheses first, but don’t forget that the fraction bar acts like parentheses, so evaluate the numerator, evaluate the denominator, and then divide.

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EXERCISE 1.6

Evaluate each expression for the given values of the variables.

1. 3x − 7 for x = 7

2. 14 − 5x for x = 6

3. x2 + 2x − 7 for x = −4

5. 3x2 − 5x + 13 for x = 2

6. x + 2y for x = 7, y = −3

7. 5x − 3y for x = 11, y = −10

8.(2x2 + 5)(4 − y) for x = −3, y = −6

10. −4x2 + 5xy − 3y2 for x = −1, y = 2

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•2•

Linear equations

Linear equations take their name from the fact that their graph is a line. They are equations that contain the first power of a variable. No exponents or radicals are involved, and no variables show up in denominators. Linear equations can be simplified to the form ax + b = 0. Examples of simplified linear equations include 3x + 7 = 0 and , but before being simplified, they might look like 4x − 9 = 2x + 1 or −5(8 − y) = 2(y + 4) − 9. If an equation has more than one variable term and one constant term on either side of the equal side, take the time to simplify it before starting to solve.

Solving a linear equation is an inverse, or undoing, process. In the equation 3x + 7 = 0, the variable x was multiplied by 3 and then 7 was added. Solving the equation involves performing opposite operations—subtraction and division—in the opposite order.

Addition and subtraction equations

If y + 4 = 7, then you can find the value of y that makes the equation true by subtracting 4 from both sides of the equation to undo the addition.

y + 4 − 4 = 7 − 4

y = 3

If the equation was formed by adding, you solve it by subtracting. If it was formed by subtracting, you solve it by adding. The equation p − 7 = 2 can be solved by adding 7 to both sides.

p − 7 + 7 = 2 + 7

p = 9

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EXERCISE 2.1

Solve each equation by adding or subtracting the appropriate number to both sides.

1. x + 8 = 12

2. y − 5 = 11

3. t + 3 = 6

4. w − 13 = 24

6. z − 2.8 = 10.3

8. x + 14 = 8

9. y − 7 = −4

10. t + 3 = − 4

Multiplication