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Math 8 Quick Study Guide. M8A1d: Solving Equations for one variable in terms of another Given the equation d = rt , you can solve for any of the variables by using the properties of equality To solve for r: d = rt d = rt (divide each side by t) so, r = d

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## M8A4f: Writing & Graphing Linear Equations: y = 5x m = 5; b = 0 y = ½ x + 1 m = ½; b = 1

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**Math 8 Quick Study Guide**M8A1d: Solving Equations for one variable in terms of another Given the equation d = rt, you can solve for any of the variables by using the properties of equality To solve for r: d = rt d = rt (divide each side by t) so, r = d ttt To solve for t: d = rt d = rt (divide each side by r) so, t = d r rr y x • M8N1d-e: +/- roots Square Root of Zero • √4 = 4, -4 or ± 4 • √0 = 0 • M8N1f: Estimating square roots • To estimate √10 • Find two consecutive integers √10 falls • between - It falls between √4 and √9 • So, the solution to estimating √10 is about 3.2 • M8N1g: Simplifying square roots To simplify √50 • List all factors of 50 - they are • 1, 2, 5, 10, 25, 50 • Identify the largest perfect square • it is 25 • 3) √50 = √25 ∙ √2 = 5 √2 • M8N1g. Adding & Subtracting square roots • ***To add square roots, the square root must be the same on each term! • Ex#1 2√3 + √3 = 3 √3 Ex #2 5 √7 – 3 √7 = 2 √7 • M8N1g: Multiplying & Dividing square roots • *** The square root do NOT have to have to be the same! • √3 ∙ √5 = √15 • √3 ∙ √3 = √9 = 3 • 2 √3 ∙ 6√5 = 12 √15 • 10 √3 = 2 • 5 √3 • M8N1h: Rational & Irrational Numbers • 0, ½ , -100, √4, √100, 0.5, 0.33 are all rational • √2, √5, √13, -√2, Π, - Π are all irrational • M8N1i: Simplifying exponents • x2∙x3 = x2 + 3 or x5 • 10x13 = 2x13-7 or 2x6 • 5x7 • 23 ∙33 = (2∙3)3 or 63 or 216 • (x2y3)2 is x4y6 • Negative exponents • 4-2 can be rewritten as1 or 1 • 42 16 • Zero exponents • ANYTHING raised to the power of zero is 1 !!! • 10,000,0000 = 1, a0 = 1, -190 = 1 • M8N1j: Scientific Notation • When you convert from a decimal number to scientific notation, the exponent will be negative Ex. 10,000,000 = 1.0 x 107 • When you convert from a whole number to scientific notation, the exponent will be positive • Ex. 0.00032 = 3.2 x 10-4[ • The number multiplied by the base of 10 raised to a power must be between 1 and 9.999 M8A4a & b Slope & y-intercept • slope – the ratio of rise over run, also known as the constant rate of change in a linear equation or the coefficient of “x” in a linear equation or the “steepness” of a line. The slope is represented by the variable “m” in a linear equation. • y-intercept – the point on a line that crosses the y-axis, also known as the “fixed amount” in a linear equation. The y-intercept is represented by the variable “b” in a linear equation. • Examples: • y = 5x m = 5; b = 0 Parallel LinesPerpendicular Lines • y = ½ x + 1 m = ½; b = 1 y = 2x and y = 2x = 1 y = 2x and y = -½x • y = 9 m = 0; b = 9 are parallel lines since are perpendicular lines • y = 3x – 6 m = 3; b = -6 they have the same slope since they have opposite • y= x m = 1; b = 0but different y-intercepts reciprocal slopes Slopes: M8A1 c & d Solving Equations & Inequalities (M8A2a&b)7x + 2 = 9x + 3 -7x -7x (subtract 7x from each side) 2 = 2x + 3 (subtract 3 from each side) - 3 - 3 -1 = 2x (divide each side by 2) 2 2 -½ = x2g + 3(g + 1) = 13 (distribute 3 to everything in parenthesis) 2g + 3(g + 1) = 13 2g + 3g + 3 = 13 (combine like terms) 5g + 3 = 13 - 3 -3 (subtract 3 from each side)5g = 10 (divide each side by 5) 5 5 g = 2 Negative Zero Positive No Slope SlopeSlopeSlope M8A2c & M8A4e Graphing Inequalities • > OPEN < CLOSED • < OPEN >CLOSED Graph x < 5 Graph x > 10 The “special rule” When you multiply or divide by a negative number, the inequality sign reverses! Example: -2x < 10 To solve, divide each side by -2 -2x < 10 -2 -2 The inequality sign will reverse x > -5 M8A4c & dConverting between slope-intercept & standard forms Standard form: Ax + By = C For example, in the equation 2x + 4y = 10 A = 2, B = 4 and C = 10 Slope-intercept form: y = mx + b m = slope and b = y-intercept To convert from standard to slope-intercept form: Example: 4x + 2y = 16 (standard form) You want to solve for y to get the equation in slope-intercept form 4x + 2y = 16 -4x -4x (subtract 4 from each side) 2y = -4x + 16 2y = -4x + 16 (divide each side by 2) 2 2 2 y = -2x + 8 (slope-intercept form) • To convert from slope-intercept to standard form: • Example: y = 3x + 1(slope-intercept form) • You want to get x and y on the same side • y = 3x + 1 • -3x -3x (subtract -3x from each side) • -3x + y = 1 (slope-intercept form) M8A4f: Writing & Graphing Linear Equations: y = 5x m = 5; b = 0 y = ½ x + 1 m = ½; b = 1 y = 9 m = 0; b = 9 y = 3x – 6 m = 3; b = -6 y= x m = 1; b = 0 To write an equation given the slope of 2and the point (1,2) 1st Use the coordinate to rewrite the equation substituting in 1 for x and 2 for y and m for the slope y = mx+ b 2 =2(1) + b 2 = 2 + b -2 -2 + b 0 = b Equation y = 2x + 0 or y = 2x Write an equation given the points (1,3) and (4, 6) 1st Plot the points 2nd Determine the slope by going from (1,3) to (4, 6) slope is 3/3 or 1 3rd Substitute the slope into the equation y = mx + b so, y = 1x +b use either point and plug into the equation 3= 1(1) + b, solve for b 2 =b Equationy = 1x +2 M8A4c: Graphing linear equations To graph a linear equation, follow these steps: Identify the slope (m) and y-intercept (b) Plot the y-intercept (0,b) Use the slope in rise to find and plot the next point run Draw a line between the two points Example: On the graph, the y-intercept is 3, so the point (0,3) is graphed The slope is 3 so, we go up by 3 and over by 2 to get the point (3,5) 2 M8A3: Relations & Functions A relation is a function if none of the x values repeat. Function : Not a function: {(1,2), (2, 4), (3, 6)} {(1, 2), (1, 3), (2,3)} {(6, 5), (4, 10), (2,15 )} {(½, ¾), (-½, ½), (0,0)} run rise (3, 5) M8A5: Systems of Equations To solve a “system” of linear equations, you need to find the point of intersection for both lines. • Elimination (cancel out one of the variables) • To solve the system 4x – y = 6 • 3x + 2y = 21 • 1.Multiply one of the equations by a number so that one of the variable will cancel out • 2. Add the equations together • 3.Use what you found for x, to find y • 1. 2(4x – y = 6) 8x – 2y = 12 Notice how the 2y and -2y CANCEL each other out • + 3x + 2y = 21 • 11x = 33 • x = 3 • Now we can use x =3 to find the value of y by substituting it back into either equation: • In 3x + 2y = 21 3(3) + 2y = 21 • 9 + 2y = 21 The solution to the system 4x – y = 6 is the point (3, 6) • -9 -9 3x – 2y = 6 • 2y = 12 • y = 6 (0, 3)**M8G1. Properties of parallel and perpendicular lines and**understand the meaning of congruence. Transversal and Parallel lines: • When a transversal line intersects parallel lines alternate interior angles are equal in measure an alternate exterior angles are equal in measure. KL is transversal Alternate Interior angles: two angle angles on opposite sides of transversal both between two lines. Pair of alternate interior angles: <3 and <5, <6 and <4 Alternate exterior angles: A pair of non-adjacent exterior angles lying on opposite sides of a transversal.Pair of alternate exterior angles: <2 and <8, <7 and < 1 Corresponding angles: two nonadjacent angles on the same side of a transversal, one between two lines and outside the lines. • Pairs of corresponding angles • :<1 and <5, <2 and <6, <3 and <7, <4 and <8 M8D1a. Set theory. A set is a collection of things. The "things" in the set are called the "elements", and are listed inside curly braces. {1, 2, 3, 4, 5, 6,} is a set. The numbers 1-6 are elements of the set. Sets can be related to each other. If one set is "inside" another set, it is called a "subset". Suppose A = {1, 2, 3} and B = {1, 2, 3, 4, 5, 6}. Then A is a subset of B, since everything in A is also in B. A union( U) of sets is the combining of two or more sets into one big set. An intersection(∩) is what two or more sets have in common. A complement is an element not contained in the set. When sets have nothing in common, they are called disjoint(Ø) sets. To show a disjoint set, you write.Set notation symbols are shorthand, but mean the same thing.Example: C = {1, 3, 5, 7, 9, 11, 13} and D = {2, 4, 6, 8, 10, 12, 14}. CUD(union of C and D) is {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}C’(the complement of C) is 2, 4, 6, 8, 10, 12, and 14. What is the intersection of sets C and D? K H G 2 1 3 4 6 5 I L J 7 8 • M8D2a. Tree Diagrams & Outcomes • Tree diagram—a diagram that illustrates all the possible outcomes of an experiment containing 2 or more independent event. Steps to make a tree diagram: 1. List all the outcomes of the first event. 2. Keep building until all outcomes are listed. • M8D2a. Addition and multiplication principles of counting • When dealing with the occurrence of more than one event, it is important to be able to quickly determine how many possible outcomes exist. • Example: if ice cream sundaes come in 5 flavors with 4 possible toppings, how many different sundaes can be made with one flavor of ice cream and one topping? Rather than list the entire sample space with all possible combinations of ice cream and toppings, we may simply multiply 5 • 4 = 20 possible sundaes. This simple multiplication process is known as the Counting Principle. The Counting Principle works for two or more activities. • M8G2. Students will understand and use the Pythagorean Theorem. • Pythagorean Theorem states that in a right triangle, the sums of the squares of the legs of the triangle are equal to the square of the hypotenuse. • Or a2 + b2 = c2 where a & b are the length of the legs and c is the length of the hypotenuse. This relationship is only true for right triangles. • Formula: a2 + b2 = c2 • 32 + 42 = c2 • 9 + 16 = c2 • 25 = c2 • √25 = √c2 • 5 = c hypotenuse leg leg • M8D3a. Basic laws of probability. • a. Find the probability of simple independent events. • Probability - the desired outcome ÷ total number of outcomes. • A single event involves the use of ONE item such as: • one person being chosen • one card being drawn • one coin being tossed • one die being rolled • Example: From a normal deck of 52 cards, what is the probability of choosing the queen of clubs? • The deck contains only one queen of clubs, so the probability will be 1/52. • M8D3b. Probability of compound independent events • A compound event involves the use of two or more items such as: • two cards being drawn • three coins being tossed • two dice being rolled • four people being chosen • If A and B are independent events, P(A and B) = P(A) x P(B). • Example: A drawer contains 3 red paperclips, 4 green paperclips, and 5 blue paperclips. One paperclip is taken from the drawer and then replaced. Another paperclip is taken from the drawer. What is the probability that the first paperclip is red and the second paperclip is blue? • Because the first paper clip is replaced, the sample space of 12 paperclips does not change from the first event to the second event. The events are independent. P(red then blue) = P(red) x P(blue) = 3/12 • 5/12 = 15/144 = 5/48. c2 c a2 a b2 b

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