# 4.1 – Patterns and Lines

Objectives

• Explore relationships of direct variation, such as diameter to circumference.
• Derive the equation of the line from the best-fit 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 xy-plane formed by a horizontal number line (the x-axis) and a vertical number line (the y-axis) that intersect at the zero point of each.
• It is also called the xy-plane or coordinate plane.
• Coordinates – The x- and y-values that describe the location of a point in an xy-plane.
• x-coordinate – The coordinate of a point that gives its distance to the right or left of the origin on a coordinate plane.
• The x-coordinate is the first number in the ordered pair that gives the point’s location.
• y-coordinate – The coordinate of a point that gives its distance above or below the origin on a coordinate plane.
• The y-coordinate 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 x-value and y-value 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 $\overleftrightarrow{PQ}$ 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 best-fit 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 $k=\frac{y}{x}$ 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: $k=\frac{y}{x}$ with the values substituted: $k=\frac{6}{3}$.
• 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: $k=\frac{y}{x}$ with the values substituted: $k=\frac{2}{1}$.
• k=2 when  you reduce the fraction, so the equation is: y=2x once again!

• Example 2
• What is the equation of a line that includes the point (2,-6) and goes through the origin (0,0) of the xy-plane?
• $k=\frac{y}{x}$ so, $k=\frac{-6}{2}$
• 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

• Example: for the line $y=\frac{2}{5}x$, if the y-coordinate is 20, what is the x-coordinate?
• Substitute 20 in for y and solve: $20=\frac{2}{5}x$
• 1. Multiply by 5 on both sides: 100 = 2x
• 2. Divide by 2 on both sides: 50 = x.