Graphing Linear Equations and Linear Systems Chapter 2 Notes Graphing Linear Equations and Linear Systems

2.1 Graphing Linear Equations A linear equation is an equation whose graph makes a line. All points on the line are solutions of the equation. To graph a linear equation: Make a table of values (an X and Y table). You can choose any values for X, but it’s easier to stay around the origin. The most common X values are -1, 0, 1, 2. Plug your X values into your equation to solve for Y. Each (X,Y) combination is one point on your graph. An equation that has just a Y is a horizontal line through that number. An equation that has just an X is a vertical line through that number.

2.2 Slope Slope-the rate of change between any two points on a line. It is basically the steepness of a line. To find the slope, find the ratio of the change in Y (the vertical change or the rise) to the change in X (the horizontal change or the run). Slope = 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑌 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑋 Slope = 𝑟𝑖𝑠𝑒 𝑟𝑢𝑛 Slope = 𝑦 2− 𝑦 1 𝑥 2 − 𝑥 1

2.3 Graphing From Slope-Intercept Form y=mx + b is called the slope-intercept form because when the equation is written like this, it gives you the slope and the y- intercept. m is the slope (rise/run) b is the y intercept (point where the line crosses the y-axis) To graph using the slope-intercept form, first plot the y-intercept (the point should be on the y-axis). From that point, use the slope to find another point on the line. Draw your line through those two points (don’t forget arrows!).

2.4 Graphing From Standard Form Standard form is ax + by = c where a and b and c are all numbers. There are two ways to graph an equation that is written in standard form. The first way is to solve for y and write the equation in slope- intercept form. Using the slope-intercept form, you plot the y- intercept first and from that point use the slope to find a second point. The second way is to keep the equation in standard form and find both the x and y intercepts . To find the x intercept, make y zero and solve for x. To find the y intercept, make x zero and solve for y.

2.5 Systems of Equations A set of two or more linear equations. The solution is an ordered pair (X,Y) that makes each equation true. There are three ways to solve a system of equations. You can make a table of values for both equations and find an X value where they both have the same Y value. You can graph each equation and find the point of intersection. You can solve it algebraically by making each expression equal to each other. First solve for X. Then plug that value into either original equation and solve for Y. Remember that your solution is a point.

2.6 Special Systems of Linear Equations A system of linear equations can have one solution, no solution, or infinite solutions. To determine the number of solutions, put each equation into slope- intercept form. If each equation has a different slope, the system of linear equations has one solution (This is the point at which they intersect). If each equation has the same slope but a different Y-intercept, the system has no solution (They are parallel and will never intersect). If each equation has both the same slope and Y-intercept, the system has infinite solutions (They are the same line!).

2.7 Solving Equations By Graphing To solve an equation where the same variable is on both sides of the equation (ax + b = cx + d), write it as two linear equations. Take whatever is to the left of the equals sign and put it in slope- intercept form (y=ax +b). Take whatever is to the right of the equals sign and also put it in slope-intercept form (y=cx + d). Graph each equation on a coordinate grid. Find the point at which they intersect. The X-value of the solution of the system of linear equations is the solution of the original equation (ax + b = cx + d).