Linear Equations in Two Variables Graphing Linear Equations

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Linear Equations in Two Variables Graphing Linear Equations Slide 2 Writing Equations and Graphing These activities introduce rates of change and defines slope of a line as the ratio of the vertical change to the horizontal change. This leads to graphing a linear equation and writing the equation of a line in three different forms. Slide 3 The Coordinate Plane Graphing ordered pairs in the coordinate plane Slide 4 The coordinate plane is formed by placing two number lines called coordinate axes so that one is horizontal (x-axis) and one is vertical (y-axis). These axes intersect at a point called the origin (0, 0), which is labeled 0. An ordered pair (x, y) is a pair of real numbers that correspond to a point in the coordinate plane. The first number in an ordered pair is the x-coordinate and the second number is the y-coordinate. Coordinate axes origin quadrants ordered pair Slide 5 In a previous lesson, you learned to plot points from a table of values. The solution to an equation in two variables is all ordered pairs of real numbers (x, y) that satisfy the equation. You can graph the solution to an equation in two variables on the coordinate plane by using the (x, y) values from the table. Graph each ordered pair in the table. Do all the points appear to lie on the line? Slide 6 Make a table of ordered pairs for each equation. Then graph the ordered pairs. (x = -3, -2, -1, 0, 1, 2, 3) 1) y = x + 3 2) y = x 2 3) y = 2x 4) y = -2x 5) y = - 2 6) y = -x + 1 7) y = x 2 1 8) y = 3 9) y = 2x 2 - 1 Slide 7 In everyday life, one quantity often depends on another. On an automobile trip, the amount of time that you spend driving depends on your speed. In cases like these, it is said that the second quantity is a function of the first. The second quantity is called dependent, while the first is independent. Can you think of other situations in real-life where one quantity is dependent on another. Slide 8 Many functions (equation) can be represented by an equation in two variables. The variable that represents the dependent quantity is called the dependent variable. The variable that represents the independent quantity is called the independent variable. When you write an ordered pair using these variables (x, y), the independent variable appears first. (independent, dependent) Slide 9 In the function y = 2x, each x-value is paired with exactly one y-value. Thus the value of y depends on the value of x. In a function, the variable of the domain is called the independent variable and the variable of the range is called the dependent variable. On a graph, the independent variable is represented on the horizontal axis and the dependent variable is represented on the vertical axis. Slide 10 Example The value v (in cents) of n nickels is given by the function (equation) v = 5n a) Identify the independent and dependent variable. b) Represent this function in a table for n = 0, 1, 2, 3, 4, 5, 6. c) Represent this function in a graph. Slide 11 Use both a table and a graph to represent each function for the given values of the independent variable. 1) The value v of 0, 1, 2, 3, 4, 5, and 6 dimes is a function of the number of dimes, d. 2) The distance d traveled by a jogger who jogs 0.1 miles per minute over 0, 1, 2, 3, 4, and 5 minutes is a function of time, t. 3) One hat cost $24. The total cost c of 1, 2, 3, 4, 5, and 6 hats is a function of the number n of hats purchased. Domain Range Try another Slide 12 What is the independent variable? What is the dependent variable? What function represents this situation? A 1200-gallon tank is empty. A valve is opened and water flows into the tank at the constant rate of 25 gallons per minute. 1) How would you find the amount of water in the tank after 5 minutes, 6 minutes, and 7 minutes? 2) How would you find the time it takes for the tank to contain 500 gallons, 600 gallons, and 700 gallons? Slide 13 Slope and Rate of Change Finding the slope of a Line Slide 14 Slope A measure of the steepness of a line on a graph; rise divided by the run. If P (x 1, y 1 ) and Q (x 2, y 2 ) lie along a nonvertical line in the coordinate plane, the line has slope m, given by Linear equation An equation whose solutions form a straight line on a coordinate plane. Collinear Points that lie on the same line. Slide 15 Remember, linear equations have constant slope. For a line on the coordinate plane, slope is the following ratio. This ratio is often referred to as rise over run. Slide 16 Slope and Orientation of a Line In the coordinate plane, a line with positive slope rises from left to right. A line with negative slope falls from left to right. A line with 0 slope is horizontal. The slope of a vertical line is undefined. Slide 17 Find the slope of the line that passes through each pair of points. 1)(1, 3) and (2, 4) 2)(0, 0) and (6, -3) 3)(2, -5) and (1, -2) 4)(3, 1) and (0, 3) 5)(-2, -8) and (1, 4) Slide 18 Finding points on a Line Graphing a line using a point and the slope is shown here. Graph the line passing through (1, 3) with slope 2. a) Graph point (1, 3). b) Because the slope is 2, you can find a second point on the line by counting up 2 units and then right 1 unit. C) Draw a line through point (1, 3) and the new point. Slide 19 A line with slope contains P(0, 5). a) Sketch the line b) Find the coordinates of a second point on the line. Slide 20 Sketch each line from the information given. Find a second point on the line. a) point H(0, 0), slope 2 b) point S(3, 3), slope -3 c) point J( 2, 5); slope 1/2 d) point T(-3, 1); slope 2/5 Slide 21 Finding Slope from a Graph Choose two points on the line (-4, 4) and (8, - 2). Count the rise over run or you can use the slope formula. Notice if you switch (x 1, y 1 ) and (x 2, y 2 ), you get the same slope: Slide 22 Real-world Applications You have been solving problems in which quantities change since early school days. For instance, here is a typical first-grade problem: If I have two apples, and then I buy five more apples, how many apples do I have in all? This situation concerns a change in the number of apples. Try this problem Slide 23 150/3 compares quantities and is called rate. One marker on a road showed 20 miles. The next marker showed 170 miles. How many miles have gone by? The time was 2:00 pm at the first marker. Then it was 5:00 pm at the second marker. How may hours have passed? As you can see, simple subtraction is all that is required to answer the questions. The answers, 150 miles and 3 hours, indicate a change in miles and a change in time, respectively. Slide 24 Since each quantity describes a change, the expression is a rate of change. You read the rate as 150 miles in 3 hours or 150 miles per 3 hours. In everyday life, most rates are given per one unit of a quantity. This is called a unit rate. So, it is more common to perform the division 150/3 = 50 to arrive at a unit rate of change: 50 miles per 1 hour or 50 miles per hour. Rates can be positive (+) or negative (-) Slide 25 After stopping to buy gas, a motorist drives at a constant rate as indicated below. Time (t) 0 hrs 2 hrs Dist (d) 5 mi120 mi Find the speed or rate of change for the car then graph the situation. Problem solving Slide 26 Calculate each rate of change 1) A motorist drives 140 miles in 3 hours and 280 miles in 4.4 hours. 2) After 5 minutes a temperature of 35 F was recorded. After 6.5 minutes, the temperature was 32 F. 3) A hose flows 250 gallons of water in 2 minutes and 875 gallons in 7 minutes. Slide 27 4) A small aircraft begins a descent to land. At 1.5 minutes the altitude is 5450 feet, and at 3 minutes the altitude is 4700 feet. If the plane descends at a constant rate, what is the rate of descent. 5) 32 people exit the stadium in 2.5 minutes and 384 people leave the stadium in 30 minutes. 6) A race car drives 2 miles in 0.02 hours and 305 miles in 3.02 hours. 7) After 3.5 minutes of descent, a small planes altitude is 3425 feet. After 6.0 minutes, the planes altitude if 2375. 8) Does the following data indicate a constant speed? Explain. 2 mi/4 min8 mi/16 min10 mi/24 min Slide 28 A linear equation is an equation whose solutions fall on a line on the coordinate plane. All solutions of a particular linear equation fall on the line, and all the points on the line are solutions of the equation. Look at the graph to the left, points (1, 3) and (-3, -5) are found on the line and are solutions to the equation. Slide 29 If an equation is linear, a constant change in the x- value produces a constant change in the y- value. The graph to the right shows an example where each time the x-value increases by 2, the y-value increases by 3. Slide 30 The equation y = 2x + 6 is a linear equation because it is the graph of a straight line and each time x increases by 1 unit, y increases by 2 Slide 31 Using Slopes and Intercepts x-intercepts and y-intercepts Slide 32 x-intercept the x-coordinate of the point where the graph of a line crosses the x-axis (where y = 0). y-intercept the y-coordinate of the point where the graph of a line crosses the y-axis (where x = 0). Slope-intercept form (of an equation) a linear equation written in the form y = mx +b, where m represents slope and b represents the y- intercept. Standard form (of an equation) an equation written in the form of Ax + By = C, where A, B, and C are real numbers, and A and B are both 0. Slide 33 Standard Form of an Equation The standard form of a linear equation, you can use the x- and y- intercepts to make a graph. The x-intercept is the x-value of the point where the line crosses. The y-intercept is the y-value of the point where the line crosses. Ax + By = C Slide 34 To graph a linear equation in standard form, you fine the x-intercept by substituting 0 for y and solving for x. Then substitute 0 for x and solve for y. 2x + 3y = 6 2x + 3(0) = 6 2x = 6 x = 3 The x-intercept is 3. (y = 0) 2x + 3y = 6 2(0) + 3y = 6 3y = 6 y = 2 The y-interce