Alg 1 (1): Solving Basic Equations

n = 4

From "Free Algebra 1 Worksheets" by Kuta Software LLC¹: 5n + 34 = −2(1 − 7n)

x = 5

From "Free Algebra 1 Worksheets" by Kuta Software LLC¹: −(1 + 7x) − 6(−7 − x) = 36

x = 4

From "Free Algebra 1 Worksheets" by Kuta Software LLC¹: −3(4x + 3) + 4(6x + 1) = 43

-17

From "Free Pre-Algebra Worksheets" by Kuta Software LLC¹:

-3

From "Free Pre-Algebra Worksheets" by Kuta Software LLC¹:

15

From "Free Pre-Algebra Worksheets" by Kuta Software LLC¹:

6

From "Free Pre-Algebra Worksheets" by Kuta Software LLC¹: 1 − r = −5

12

From "Free Pre-Algebra Worksheets" by Kuta Software LLC¹: x/(−4) − 5 = −8

8

From "Free Pre-Algebra Worksheets" by Kuta Software LLC¹: −9 + n/4 = −7

E) 10x = 3y

From The College Panda ACT Math: Advanced Guide and Workbook (p. 15)²: If x = 1/2z and 3y = 5z, which of the following relationships holds between x and y for each nonzero value of z? A) 2x = 15y B) 3x = 10y C) 5x = 6y D) 6x = 5y E) 10x = 3y

C) x = (1 + y)²

From The College Panda ACT Math: Advanced Guide and Workbook (p. 15)²: Whenever √x - y = 1 for positive values of x, which of the following equations gives x in terms of y? A) x = 1 - y² B) x = 1 + y² C) x = (1 + y)² D) x = (1 - y)² E) x = y² - 1

-6

From The College Panda ACT Math: Advanced Guide and Workbook (p. 17)²: If (x + 3)² = 121, what is the sum of the two possible values of x?

8 + 1/2, or 17/2

From The College Panda ACT Math: Advanced Guide and Workbook (p. 17)²: x = ?

2q / (q - 2)

From The College Panda ACT Math: Advanced Guide and Workbook (p. 18)²: Find p in terms of q

3/2

From The College Panda ACT Math: Advanced Guide and Workbook (p. 19)²:

(b - c)/a

From The College Panda ACT Math: Advanced Guide and Workbook (p. 19)²: For al nonzero values of a, b, and c, what is the solution for x of the equation b - ax = c?

-0.87

From The College Panda ACT Math: Advanced Guide and Workbook (p. 19)²: For what value of x is the equation 5.7x + 8.85 = 1.02 - 3.3x true?

-1

From The College Panda ACT Math: Advanced Guide and Workbook (p. 19)²: If 7 + (x - 7) = 7 - (7 + y) and y ≠ 0, what is the value of x/y?

B) 8

From The College Panda ACT Math: Advanced Guide and Workbook (p. 19)²: If x and y are positive integers and x - y = 7, what is the least possible value of xy? A) 7 B) 8 C) 9 D) 12 E) 18

B) 8

From The College Panda ACT Math: Advanced Guide and Workbook (p. 19)²: What is the largest value of a for which there exists a real value of b such that a² + b² = 64? A) 6 B) 8 C) 10 D) 32 E) 64

3/2

From The College Panda ACT Math: Advanced Guide and Workbook (p. 19)²: What value of a will make the following equation true?

10/9

From The College Panda ACT Math: Advanced Guide and Workbook (p. 20)²:

A) z < x < y

From The College Panda ACT Math: Advanced Guide and Workbook (p. 20)²: For positive real numbers x, y, and z such that 5x = 2y and 1/6y = 2/3z, which of the following inequalities is true? A) z < x < y B) x < z < y C) z < y < x D) x < y < z E) y < x < z

C) -x - y + 7

From The College Panda ACT Math: Advanced Guide and Workbook (p. 25)²: Given the equations M = 3 - x and y = N + 4, which of the following expressions is equivalent to M - N written in terms of x and y? A) -x - y - 1 B) -x - y + 1 C) -x - y + 7 D) -x + y - 1 E) -x + y + 7

a = 4b 1) Cross multiply (a - b) / (a + b) = 3/5 5(a - b) = 3(a + b) 2) Distribute 5(a - b) = 3(a + b) 5a - 5b = 3a + 3b 3) Combine like terms and solve 5a - 5b + 5b = 3a + 3b + 5b 5a - 3a = 3a - 3a + 8b 2a ÷ 2 = 8b ÷ 2 a = 4b

How do you isolate a variable if it's trapped in multiple parts of a fraction? Use your knowledge to evaluate the following from The College Panda ACT Math: Advanced Guide and Workbook (p. 17)²: What is a in terms of b?

k = -1 1) If there are parentheses, use the distributive property to dissolve them −18 − 6k = 6(1 + 3k) −18 − 6k = 6 + 18k 2) Combine like terms and solve −18 − 6k + 6k = 6 + 18k + 6k -18 - 6 = 6 - 6 + 24k -24 ÷ 24 = 24k ÷ 24 -1 = k

How do you solve an equation if the variable appears in multiple locations? Use your knowledge to solve the following equation from "Free Algebra 1 Worksheets" by Kuta Software LLC¹: −18 − 6k = 6(1 + 3k)

x = 10 / 3 When you see a fraction in an equation, you can move it by multiplying both sides by the reciprocal (3/5)x = 2 *5/3 *5/3 x = 10 / 3

How do you solve an equation like: 3/5x = 2

x = 7 To isolate x, always do the opposite of the number next to it. 5x = 35 The opposite of "× 5" is "÷ 5," so we ÷ 5 on both sides 5x = 35 5x ÷ 5 = 35 ÷ 5 x = 7

How do you solve an equation like: 5x = 35

x = 4 To isolate x, always do the opposite of the number next to it. x + 6 = 10 The opposite of "+ 6" is "- 6," so we - 6 from both sides x + 6 - 6 = 10 - 6 x = 4

How do you solve an equation like: x + 6 = 10

x = 7 To isolate x, always do the opposite of the number next to it. x - 5 = 2 The opposite of "- 5" is "+ 5," so we +5 to both sides x - 5 + 5 = 2 + 5 x = 7

How do you solve an equation like: x - 5 = 2

x = 32 To isolate x, always do the opposite of the number next to it. x/4 = 8 The opposite of "/4" is "× 4," so we × 4 on both sides x/4 = 8 x/4 × 4 = 8 × 4 x = 32

How do you solve an equation like: x/4 = 8

x = 59 1) Start by isolating the radical on one side using PEMDAS backwards 4√(x - 10) + 15 = 43 4√(x - 10) = 28 (Subtraction) √(x - 10) = 7 (Division) 2) Square both sides and solve for x √(x - 10)² = 7² x - 10 = 49 x = 59 (Addition)

How do you solve an equation with a variable inside a root? Use your knowledge to evaluate the following: 4√(x - 10) + 15 = 43

x = 59 1) Start by isolating the exponential expression on one side using PEMDAS backwards 4(x - 5)² + 15 = 115 4(x - 5)² = 100 (Subtraction) (x - 5)² = 25 (Division) 2) Take the square root of both sides and solve for x √(x - 5)² = ±√25 x - 5 = ±5 x = ±5 + 5 (Addition) x = 0, 10

How do you solve an equation with a variable raised to an exponent? Use your knowledge to evaluate the following: 4(x - 5)² + 15 = 115

-4 1) Dissolve the fraction first by multiplying the denominator on both sides (k - 10) / 2 × 2 = -7 × 2 k - 10 = -14 2) Solve normally k - 10 + 10 = -14 + 10 k = -4

How do you solve an equation with expressions embedded in a fraction? Use your knowledge to solve the following equation from "Free Pre-Algebra Worksheets" by Kuta Software LLC¹:

r = 10 In a multi-step equation, follow PEMDAS backwards to solve for the variable. Parentheses Exponents Multiplication Division Addition Subtraction r/10 + 4 = 5 1) Check for numbers that can be added or subtracted to the other side. They must be by themselves, and not embedded in a fraction! In this case, we can subtract 4 to the other side r/10 + 4 - 4 = 5 - 4 r/10 = 1 2) Check for numbers that can be multiplied or divided to the other side. In this case, we can multiply 10 on both sides r/10 × 10 = 1 × 10 r = 10

How do you solve an equation with multiple steps? Use your knowledge to solve the following question from "Free Pre-Algebra Worksheets" by Kuta Software LLC¹: r/10 + 4 = 5

x = ±4 When there's an exponent, take the root of both sides √x² = √16 x = ±4 *The even root of any number is ±*

How do you solve x² = 16?

x = 2 When there's an exponent, take the root of both sides ₃√x³ = ₃√8 x = 2

How do you solve x³ = 8?

x = 8 When there's a root, raise both sides to the root number (₃√x)³=2³ x = 8

How do you solve ₃√x = 2?

x = 36 Recall: Square roots and ² cancel out (√x)²=6² x = 36

How do you solve √x = 6?