Objectives
 Explore relationships of direct variation, such as diameter to circumference.
 Derive the equation of the line from the bestfit line.
 Construct a linear equation from an English sentence
 Graph a linear equation from a chart of solutions to the equation.
Key Terms
 Cartesian Coordinate System – An xyplane formed by a horizontal number line (the xaxis) and a vertical number line (the yaxis) that intersect at the zero point of each.
 It is also called the xyplane or coordinate plane.
 Coordinates – The x and yvalues that describe the location of a point in an xyplane.
 xcoordinate – The coordinate of a point that gives its distance to the right or left of the origin on a coordinate plane.
 The xcoordinate is the first number in the ordered pair that gives the point’s location.
 ycoordinate – The coordinate of a point that gives its distance above or below the origin on a coordinate plane.

 The ycoordinate is the second number in the ordered pair that gives the point’s location.
 Direct variation – A relationship in which one variable (quantity) is a constant multiple of the other variable (quantity).
 You can say that y varies directly with x and write y = kx, where k is a constant.
 The relationship of direct variation is plotted on a graph as a straight line.
 The graph passes through the origin.
km traveled (y) varies directly with Hours driven (x). k = 60 (the constant speed).
 Equation of the Line – Equation that shows the relationship between the xvalue and yvalue of every point on the line.
 Graphs of Linear Equations – The set of all points on a line whose coordinates can be substituted into an equation and keep that equation true.
 Line – A straight path that continues forever in both directions.
 The symbol represents a line that contains points P and Q.
 Line of Best Fit – A line drawn as near as possible to the points in a scatterplot.
 The bestfit line helps you see the relationship shown by the scatterplot.
 Linear Equation – Any equation whose graph is a straight line.
 A linear equation can be written in the form y = mx + b.
 Origin – The point where the axes intersect in a Cartesian coordinate system.
 The coordinates of the origin are (0, 0).
Review
 Circumference and Diameter
 Circumference is the distance around a circle.
 Diameter is the distance across the center of a circle.
 The relationship between the circumference and the diameter is a constant value: π (pi), which is about 3.14 units.
 Constant values are values that never change.
 The relationship between circumference and diameter can be written as: y = 3.14x, where 3.14 is the constant.
 English: The circumference of the wheel depends on its diameter.
 Algebra: The circumference of the wheel “is a fuction of” its diameter.
 Multiples
 A multiple is the product of any two whole numbers.
 Examples
 21 is a multiple of 3 because 3 7 = 21.
 100 is a multiple of 10 because 10 10 = 100.
 Kilometers
 Units of length in the metric system of measurement.
 There are 1000 meters in 1 kilometer.
Notes
 You can use constant values to find multiples of numbers.
 Constant = 4, so, 1(4)=4, 2(4)=8, 3(4)=12, 4(4)=16, and so on!
 4, 8, 12, and 16 are all multiples of 4!
 Linear Equations
 A linear equation can be written in the form y = mx + b.
 y = 12x is a linear equation.
 So, in a linear equation, b can equal zero.
 Direct Variation is a Linear Equation: y=kx
 You can rewrite the equation as by dividing x on both sides.
 k is a constant and will never change.
 y=kx is a literal equation, as there are 2 variables: y and x.
 Constant Values (k) or (m) represent a linear equation’s slope
 The larger the value, the steeper the slope (line goes almost straight up from left to right)
 y = 10x
 The smaller the value, the less steep the slope (the line hardly goes up at all from left to right)
 y = 0.1x
 Positive and Negative Constants
 Lines that go up from left to right have a positive constant (k) value.
 Lines that go down from left to right have a negative constant (k) value.
 To find the constant (k) value, when given a graph of a line with points plotted:
 Change y=kx to k=y÷x by dividing x on both sides.
 Then, substitute in the x and y point values for ONE point and solve for k.
 Lastly, reduce the fraction and write the equation in the form y=kx, substituting the answer for k.
 Example 1
 Points: (3,6) and (1,2) are both plotted on the line in the graph below.
 You can use EITHER point, as they connect to form a straight line. All straight lines have a constant value (k).
 For (3,6): If x=3 and y=6, then use the formula: with the values substituted: .
 k=2 when you reduce this fraction, so the equation is: y=2x
 For (1,2): If x=1 and y=2, then use the formula: with the values substituted: .
 k=2 when you reduce the fraction, so the equation is: y=2x once again!
 Points: (3,6) and (1,2) are both plotted on the line in the graph below.
 Example 2
 What is the equation of a line that includes the point (2,6) and goes through the origin (0,0) of the xyplane?
 so,
 k = 3
 The equation will be y = 3x
 k is negative, which means the line goes down from left to right on the graph.
 Example: Pizza
 You measure straight across your pizza.
 It has a diameter of 22 centimeters.
 We can use the linear equation y = 3.14x to find the circumference of the pizza!
 Example: Changing Feet to Inches
 There are 12 inches in a foot, so y = 12x
 y: inches
 x: feet
 12: constant
 There are 12 inches in a foot, so y = 12x
 Example: for the line , if the ycoordinate is 20, what is the xcoordinate?
 Substitute 20 in for y and solve:
 1. Multiply by 5 on both sides: 100 = 2x
 2. Divide by 2 on both sides: 50 = x.
 Substitute 20 in for y and solve: