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Properties of equalities: A1.1.2. 1.2 Use, identify algebraic property to justify steps in an equation solving process Properties of equalities: Distributive property—multiply each item in the parenthesis by the item outside the parenthesis 3(x + 4) = 3x +12 Associative property—changing grouping 3 + (4 + 5) = (3 + 4) + 5 Commutative property—changing order 3 + 4 = 4 + 3 Primary RULE of equalities Whatever you do to one side of the equation you must do to the other

Properties of equalities: A1.1.2. 1.2 Use, identify algebraic property to justify steps in an equation solving process Properties of equalities: Additive inverse—opposites Move entire terms from one side to the other Ex. 3x + 4 = 10 - 4 - 4 3x = 6 Multiplicative inverse—reciprocals (division) Removes the coefficient of the variable Ex. 3x = 6 3 3 x = 2

Name the properties being used: To solve the equation: A1.1.2. 1.2 Use, identify algebraic property to justify steps in an equation solving process Name the properties being used: To solve the equation: 3 ( 7 – 2x) = 14 – 8(x – 1) 21 – 6x = 14 – 8x + 8 21 – 6x = 6 – 8x 8x – 6x = 22 + (-21) 2x = 1 x = ½ Distributive property comm property & add like terms Additive inverses Add like terms Multiplicative inverse

One step of the problem is: -5 ( 3x + 7) = 10 The next is A1.1.2. 1.2 Use, identify algebraic property to justify steps in an equation solving process Typical question: One step of the problem is: -5 ( 3x + 7) = 10 The next is -15x + 35 = 10 What operation is was used? Then four choices are given.

Slope—other names Constant of Variation Rate of Change Change in Y divided by Change in X Usually represented by letter “m” in an equation

Slope Refers to the steepness of a line VIPs about slopes: The > the abs. value of the slope the steeper it is VIPs about slopes: Positive slopes Negative slopes Slope means rise +count up/-down then run right! run Examples in real life: handicap ramps, pitch of roof, where else??

Formula to find the constant of variation Use this formula when you know two points on your line.

Example Find the slope of the line that goes through the points (-5, 3) and (-4, -2).

He rented the car for 3 days and drove 150 miles. A1.1.2. 1.3 Interpret solutions to problems in the context of the problem situation (linear) James rents a car for $45 a day(x) plus $.20 a mile (y). He spent $165 renting the car. The equation below represents the relationship between the number of days a car is rented and the number of miles driven during this time. 45x + .20y = 165 The ordered pair (3, 150) is a solution. What does the solution (3, 150) represent? He rented the car for 3 days and drove 150 miles.

Given: 5x + 2y = 18 2x + y = 8 Solve by graphing: A1.1.2. 2.1 Write/solve a system of linear equations through graphing , substitution, and elimination. (limited to 2 linear equations) Given: 5x + 2y = 18 2x + y = 8 Solve by graphing: 2y = 18 – 5x y = 9 – 5/2 x y = 8 – 2x

Solve by substitution: y = 8 – 2x 5x + 2(8 – 2x) = 18 A1.1.2. 2.1 Write/solve a system of linear equations through graphing , substitution, and elimination. (limited to 2 linear equations) Given: 5x + 2y = 18 2x + y = 8 Solve by substitution: y = 8 – 2x 5x + 2(8 – 2x) = 18 5x + 16 – 4x = 18 x = 2 2x + y = 8 2(2) + y = 8 4 + y = 8 y = 4 Solution: (2, 4)

Solve by linear combination/elimination method: -10 x – 4y = -36 A1.1.2. 2.1 Write/solve a system of linear equations through graphing , substitution, and elimination. (limited to 2 linear equations) Given: 5x + 2y = 18 2x + y = 8 Solve by linear combination/elimination method: -10 x – 4y = -36 10 x + 5y = 40 y = 4 To eliminate the x: -2 (5x + 2y = 18) 5 (2x + y = 8 ) 2x + y = 8 2x + 4 = 8 2x = 4 x = 2 Solution: (2, 4)

A1.1.2. 2.2 Interpret solutions to problems in context of situation (2 linear equations) A delivery truck arrives at Roberts’ store with 8 small boxes and 5 large boxes. The total charge for the boxes, without tax or delivery charges, is $184. A large box costs $3 more than a small box. What is the cost of each size box. Follow the ROPES Organize: Let x = small y = large Plan: 8x + 5y = 184 y = x + 3 Which option for solving would be best? Evaluate: Use substitution 8x + 5y = 184 y = x + 3 8x + 5(x + 3) = 184 8x + 5x + 15 = 184 13x + 15 = 184 13x = 169 x = 13 y = x + 3 y = 13 + 3 y = 16 Sythesize (i.e. answer the question): Small boxes cost $13 and large boxes cost $16.

Example 2

Which of the following are linear and why? The standard form of a line is ax + by = c Ex. xy = 4 is not linear because x & y multiplied together Ex. x2 + y = 4 is not linear because exponent > 1 Ex. 3x + 2y = -7 is linear actually written in standard form

Slope Intercept Form of a Line y = mx + b m = the slope of the line b = the y-intercept of the line or where the line crosses the y-axis.

Example 3 Given the equation y = -5x + ½, identify the slope and y-intercept and describe what they both represent. Answer: The slope is always the number in front of x so the slope is -5. Since it is a negative number the line must go down to the right. The y-intercept of ½ which means the line crosses the y-axis at ½ or at the point (0, ½).

Standard Form vs. Slope-Intercept Form If you are given an equation that isn’t in slope-intercept form, solve for y so that it is. Ex. Identify the slope in the following equation. 2x + 3y = 6

Example 4 Paula catches 2 fish per day during a family fishing trip. Write an equation that shows the relationship between the length of fishing trip (d) and the total number of fish caught (f). Answer: f = 2d + 0 2 fish per day and none before they arrived

Finding equations for lines Given the slope (m) and the y-intercept b Use the slope intercept form: Y = mx + b Ex. If the slope = ½ and the y –intercept is -3 the equation of the line is: y = 1/2x – 3

Example Sometimes the keystones will challenge you to recognize simple problems in a unique format: What is the y-intercept: What is the slope:

Example – Slope Intercept Form Find the equation for a line that has a slope of -½ and goes through the point (2, 6). Therefore m = -½ the point (2, 6) (x, y) y = mx + b 6 = -½(2) + b 6 = -1 + b 7 = b EQUATION IS y = -½x + 7

Point Slope Formula y – y1 = m(x – x1) You need a point (x1, y1) and the slope. After substituting, solve the equation for y to place it in slope—intercept form. Ex. Find the line where m = 2 and containing the point (3, -4)

Example – Point Slope Form Find the equation of the line that has a slope of -½ and goes through the point (2, 6) using point slope form, convert to slope intercept form to graph m = -½ the point (2, 6) (x1, y1) y – y1 = m(x – x1) y – 6 = -½(x – 2) y – 6 = -½x + 1 + 6 + 6 y = -½x + 7

Practice Use point slope form to find Find the equation for the line that goes through the points (-10, 3)(-4, 6).

Tips for answering MC questions when given the graph: Look for the y-intercept first as you will probably be able to rule out some choices. Look at the slope to see if it is positive or negative. Can always try to count out the slope. Can always pick 2 points on your line, find the slope then use y = mx + b to write the equation! Convert all forms to y = mx + b because it is easy to pick out the slope and y-intercept from this form.

Ex. 5 A graph of a linear equation is shown:

Example 6 A juice machine dispenses the same amount of juice into a cup each time the machine is used. The equation x + 12y = 240 describes the relationship between the number of cups (x) into which juice is dispensed and the gallons of juice (y) remaining in the machine. How many gallons of juice are in the machine when it is full? A. 12 B. 20 C. 228 D. 240