Slope of a Line Unit 7 Review of Slope and Graphing Linear Equations

What is slope? In this course, we defined slope to be: A measurement of the steepness and direction of a line. Slope is represented by the letter m.

What is slope? In physics, you used the formula: In Algebra, we use the formula: Slope=change in y change in x =vertical change horizontal change

Reading Slope Slope of a line can tell us many things A flat line has a small slope, most likely a fraction! Because this line goes down (from left to right), it has a negative slope. A steeper line has a larger slope. This line has a positive slope– it goes up from left to right.

Finding Slope Because it only take 2 points to graph a line, we can find the slope given 2 points. Method 1: Slope Triangle 1. Plot the points on an x-y plane. 2. Connect the points with a straight line. 3. Draw a triangle connecting the points. 4. Use the triangle to identify the vertical and horizontal change. 1 3 The slope is (2, -3) (-1, -2)

Finding Slope Method 2: Formula Because slope is a measurement of vertical change vs. horizontal change, we can use our coordinates in a formula to find slope. Example: (2, -3) (-1, -2) The slope is

Equations and Slope To find slope from an equation, you first must be sure the equation is in y-form! Example 1: Because the equation is solved for y (y is by itself), we can quickly see the slope at the coefficient of x: The slope is Example 2: This equation is not in y form. We must first solve for y! Again, the slope is the coefficient of x, or ½.

Equations and Slope You can use slope to graph the equation of a line. * Remember, the equation of a line contains only y and x. There are no exponents! The easiest form of an equation to use is y-form, also known as SLOPE- INTERCEPT FORM. The general form is: y = mx + b When an equation is in this form, m, or the coefficient of x, is always the slope. The constant, b, is the y- intercept.

Equations and Slope Because these are linear equations, the graphs are lines. We know that we only need 2 points to graph a line. We can use the ideas of slope and y-intercept to draw the graph of an equation. The y-intercept provides us with one point that is on our graph. The slope can show us how to move to a second point. The slope is NOT a point. It is a movement to a point. Remember, slope is an amount of vertical and horizontal change, not an x and y coordinate.

Equations and Slope An Example This equation is already in y-form, or slope- intercept form! We can quickly identify the information we need to draw our graph. 1) Plot the y- intercept. (With a y- intercept of 2, we plot the point (0, 2). 2)Use the slope to move to the next point. A slope of –3 is a movement down 3 units, right 1 unit. How would you move if the slope was positive?

Equations and Slope Example 2: 2x – 4y = 8 To be able to graph a line from the equation, the equation must be in y-form. We begin this example by solving for y. Using the same steps as the previous example, we can graph the line.

Systems of Equations A system of equations is two equations in the same problem. The solution is the x and y values that make BOTH equations true. There are two methods used to solve systems: graphing and substitution. METHOD 1: Graphing– Because there are 2 equations, your graph should contain 2 lines. The solution is the point of intersection of the two lines. Graphing is not the most accurate method to solve systems. It is a useful check – step!

Systems of Equations Example 1 1) Find the slope and y- intercept for each line. 2) Graph both equations on the same set of axes. 3) The point of intersection is the solution of the system. The solution is (-3, -2).

Systems of Equations An Algebraic Solution Example 1 To solve for x and y, you have to write 1 equation with only 1 variable. To write the equation, you must replace y with a value that is equivalent! Solve the new equation for x. To find the value for y, substitute x into either of the original equations. (-3, -2) Note the solution is the same as graphing!

Systems of Equations An Algebraic Solution Example 2 Like the last example, we must use substitution to solve. In this case, we can replace y in our second equation with the equivalent expression, x + 1 Solve the equation for x. Don’t forget to find y! (-2, -1)

Review Solve each of the following systems algebraically. Then graph the equations to check your solutions. Once you have solved and graphed the systems. Click your mouse to check your solutions.