Introduction to Linear Equations Linear Equations, Tables of Values, Intercepts, and Slope

What is a line and how can it be graphed?

Linear Equation An equation for which the graph is a line. In linear equations, a constant difference in x-values results in a constant difference in y-values (straight line with a constant rise or fall) Linear equations are usually written in 1 of 2 forms Slope-intercept form y = mx + b Standard form Ax + By = C (Where A, B, C are integers)

The equations must meet the following conditions to be linear… How can we determine if an equation represents a line without graphing? (What does the equation look like?) The equations must meet the following conditions to be linear… No more than 2 variables – cannot have 3 or more different variables. No exponents on the variables. No products of variables – variables cannot be multiplied together No variables in the denominator of a fraction.

Determine if the following are linear, if not give a reason x = 6 2. y + 4 = x 3. y = 1/x 4. x2 + 3 = y 5.3x + 2y = 4 6. 4xyz = 2

How do I determine if a point lies on the line without graphing How do I determine if a point lies on the line without graphing? (Basically if it is a solution) A given point lies on a line, if and only if, when the coordinates for x and y are substituted into the equation and makes the equation true. Example: Does (2, - 4) lie on 4x = -y + 4? 4(2) = -(-4) + 4 8 = 4 + 4 8 = 8 yes it lies on the line

Determine if the given points lie on the line. Does (3, 2) lie on 2x + 3y = 12? Does (-3, 1) lie on y = 4x – 3?

How do I graph a linear equation? (Several methods) Method 1: Table of Values Steps: Solve the equation for y. (Isolate y) Create your table (See the board) Pick 5 values (2+, 2-, and zero) for x Plug x-values in to equation and evaluate for y Plot the points – draw a line through the points.

Hints for table of values: If the line is not straight; first double check your graphing of the points, did you go left/right first then up/down? Then double check your math in the table. If there is a fraction multiplier of x when solved for y, pick numbers that are multiples of the denominator, will eliminate most fractions.

Graph using a table of Values Try These Graph using a table of Values 1) y = x + 3 2) y – x = - 4

How do I graph a linear equation? (Continued) Method 2: Graphing from Intercepts What are intercepts? Intercepts are points at which the line crosses the axes. Most lines have 2, but some special lines only have one. The intercepts are called the x-intercept and the y-intercept.

X-intercept Where the line crosses the x-axis, all along the x-axis the y-value is zero! So coordinates of a x-intercept are (x,0)

Y-intercept Where the line crosses the y-axis, all along the y-axis the x-value is zero. So the coordinates for the y-intercept are (0, y).

Calculating the X-intercept Since the y-coordinate in every x-intercept is zero, we can use that to find the x-value. Plug 0 in for y and solve the equation for x. Coordinates become (x, 0)

Calculating the Y-intercept Since the x-coordinate in every y-intercept is zero, we can use that to find the y-value. Plug 0 in for x and solve the equation for y. Coordinates become (0, y)

How do I graph a linear equation? (Continued) Method 2: Graphing from Intercepts Steps: Calculate the x and y-intercepts. Plot the two intercepts. Draw a line through the two points.

Try These! ½ x + 4y = -4 2x = 3y - 6

Slope What is slope? Slope is the rise or fall of the line; basically the steepness of it It is constant over the entire length of the line

Slope Equal to: Rise Run

The change vertically, the change in y Rise The change vertically, the change in y

The change horizontally or the change in x Run The change horizontally or the change in x

4 Types of Slope Positive – a line that rises from left to right (uphill) Negative – a line that drops from left to right (downhill) 0 slope – a line that does not rise at all (horizontal) Undefined slope – a line that has no run at all (vertical)

Determining Slope Graphically – finding the slope from a graph Count the rise or vertical distance between two points Count the run or horizontal distance between the same two points Put rise run Determine if positive or negative

Determining Slope (cont) 2. Algebraically – finding slope using a formula, no graph needed Find 2 points on a line (2, 3) (5, 4) (x1, y1) (x2, y2) Plug the values into the following equation: Y2 - Y1 = 4 – 3 = 1 X2 - X1 5 - 2 3

Try These! (2, 1) and (-6, -1) (5, -3) and (2, 5)