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## Graphing Linear Equations

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**Chapter 4**Graphing Linear Equations**Chapter Sections**4.1 – The Cartesian Coordinate System and Linear Equations in Two Variables 4.2 – Graphing Linear Equations 4.3 – Slope of a Line 4.4 – Slope-Intercept and Point-Slope Forms of a Linear Equation**The Cartesian Coordinate System and Linear Equations in Two**Variables § 4.1**Definitions**A graphshows the relationship between two variables in an equation. The Cartesian (rectangular) coordinate systemis a grid system used to draw graphs. It is named after its developer, René Descartes (1596-1650).**Definitions**y x II I III IV The two intersecting axis form four quadrants,numbered I through IV. The horizontal axis is called the x-axis. The vertical axis is called the y-axis.**Definitions**y x Origin (0, 0) The point of intersection of the two axes is called the origin. The coordinates, or the value of the x and the value of the y determines the point. This is also called an orderedpair.**Plotting Points**Starting at the origin, move 3 places to the right. Plot the point (3, –4). The x-coordinate is 3 and the y-coordinate is –4.**Plotting Points**Then move 4 places down. Plot the point (3, –4). The x-coordinate is 3 and the y-coordinate is –4.**Plotting Points**(3, -4) Plot the point (3, –4). The x-coordinate is 3 and the y-coordinate is –4.**Linear Equations**A linear equation in two variablesis an equation that can be put in the form ax + by = c where a, b, and c are real numbers. This is called the standard formof an equation. Examples: 4x – 3y = 12 x + 2y = – 35**Solutions to Equations**The solutionto an equation is the ordered pair that can be substituted into the equation without changing the “validity” of the equation. Is (3, 0) a solution to the equation 4x – 3y = 12? 4x – 3y = 12 4(3) – 3(0) = 12 12 – 0 = 12 12 = 12 Yes, it is a solution.**Graphing**A graphof an equation is an illustration of a set of points whose coordinates satisfy the equation. A set of points that are in a straight line are collinear. The points (–1, 4), (1, 1) and (4, –3) are collinear.**Graph by Plotting Points**• Solve the linear equation for the variable y. • Select a value for the variable x. Substitute this value in the equation for x and find the corresponding value of y. Record the ordered pair (x,y). • Repeat step 2 with two different values of x. This will give you two additional ordered pairs. • Plot the three ordered pairs. • Draw a straight line through the points.**Graph by Plotting Points**Graph the equation y = –x + 3. y = –2 + 3 y = 1 Let x = 2 This give us the point (2, 1). y = –(-2) + 3 y = 5 Let x = -2 This give us the point (-2, 5). y = –1 + 3 y = 2 Let x = 1 This give us the point (1, 2). Plot the points and draw the line.**Graph by Plotting Points**Plot the points (2, 1), (-2, 5), and (1, 2). Draw the line.**Graph Using Intercepts**• Find the y-intercept by setting x in the equation equal to 0 and finding the corresponding value of y. • Find the x-intercept by setting the y in the equation equal to 0 and finding the corresponding value of x. • Determine a check point by selecting a nonzero value for x and finding the corresponding y. • Plot the two intercepts and the check point. • Draw a straight line through the points.**Graph Using Intercepts**y = This gives us the point (2, ). Graph the equation -3y –2x = -6. -3y –2(0) = -6 y = 2 Let x = 0 This gives us the y-intercept (0, 2). -3(0) –2x = -6 x = 3 Let y = 0 This gives us the x-intercept (3, 0). -3y – 2(2) = -6 -3y = -2 Let x = 2 Plot the points and draw the line.**Graph by Plotting Points**Plot the points (0, 2), (3, 0), and (2, ). Draw the line.**Slope of a Line**§ 4.3**Slope**The slope of a lineis the ratio of the vertical change between any two selected points on the line. Consider the points (3, 6) and (1,2).**Slope**Vertical Change Horizontal Change (3, 6) and (1,2) This means the graph is moving up 4 and to the right 2.**Slope**Simplifying, , so m = 2 Vertical Change Horizontal Change m = 2**Positive & Negative Slopes**y y x x Positive Slope Negative Slope Line rises from left to right Line falls from left to right**Horizontal Lines**x = 2 Every horizontal like has a slope of 0.**Vertical Lines**y = -4 The slope of any vertical line is undefined.**Parallel Lines**Two non-vertical lines with the same slope and different y-intercepts are parallel . Any two vertical lines are parallel to each other. m1 = m2**Perpendicular Lines**m1 = -1 m2 Two lines whose slopes are negative reciprocals of each other are perpendicular lines. Any vertical line is perpendicular to any horizontal line.**Slope-Intercept Form**slope y-intercept y-intercept is (0, -4) slope is 3 y-intercept is (0, ) slope is In the slope-intercept form, the graph of a linear equation will always be a straight line in the form y = mx+ bwere m is the slope of The line and b is the y-intercept (0, b). y = mx+ b Examples: y = 3x– 4 y = x+ **Slope-Intercept Form**y-intercept is (0, -) slope is Write the equation 6x = 8y + 7 in slope-intercept form. Solve for y. 6x = 8y + 7 – 8y = – 6x +7 y = x – y = x –**Point-Slope Form**When the slope and a point on the line are known, we can use thepoint-slope form to determine the line. where m is the slope of the line and (x1, y1) is a point. Example: point (2, 3) and slope = 4: