Understanding Linear Equations: Graphing and Examples
Learn about linear equations, collinear points, and slope in this comprehensive guide. Explore how to write, graph, and determine if equations are linear using examples. Understand the slope-intercept form and its importance in graphing.
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Writing and Graphing Linear Equations
Linear equation An equation whose solutions form a straight line on a coordinate plane. Collinear Points that lie on the same line. Slope A measure of the steepness of a line on a graph; rise divided by the run.
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.
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 X Y = 2x + 6 Y (x, y) 1 2 3 4 5 2(1) + 6 2(2) + 6 2(3) + 6 2(4) + 6 2(5) + 6 8 10 12 14 16 (1, 8) (2, 10) (3, 12) (4, 14) (5, 16)
Complete the table below, then graph and tell whether it is linear. x -2 -1 0 1 2 y = 2x + 3 y (x, y)
Can you determine if the equation is linear? The equation y = 2x + 3 is a linear equation because it is the graph of a straight line. Each time x increases by 1 unit, y increases by 2. X -2 -1 0 1 2 y = 2x + 3 2 (-2) + 3 2(-1) + 3 2(0) + 3 2(1) + 3 2(2) + 3 Y -1 1 3 5 7 (x,y) (-2, 1) (-1, 1) (0, 3) (1, 5) (2, 7)
Slope-intercept Form y = mx + b
Slope-intercept Form An equation whose graph is a straight line is a linear equation. Y = mx + b (if you know the slope and where the line crosses the y-axis, use this form) m = slope b = y-intercept
y = 3x + 6 y = 4/5x -7 m = 3 b = +6 m = 4/5 b = -7 Please note that in the slope-intercept formula; the y term is all by itself on the left side of the equation. That is very important!
WHY? If the y is not all by itself, then we must first use the rules of algebra to isolate the y term. For example in the equation: 2y = 8x + 10 Only now can we determine that: slope = 4 y-intercept = 5
OKgetting back to the lesson Your job is to write the equation of a line after you are given the slope and y-intercept Let s try one Given m (the slope remember!) = 2 And b (the y-intercept) = +9 y = mx + b y = 2x + 9
Lets do a couple more to make sure you are expert at this. Given m = 2/3, b = -12, Write the equation of a line in slope-intercept form. Y = mx + b Y = 2/3x 12
Using slope-intercept form to find slopes and y-intercepts The graph at the right shows the equation of a line both in standard form and slope-intercept form. You must rewrite the equation 6x 3y = 12 in slope-intercept to be able to identify the slope and y- intercept.
Using slope-intercept form to write equations, Rewrite the equation solving for y = to determine the slope and y-intercept. 3x y = 14 -y = -3x + 14 -1 -1 -1 y = 3x 14 or 3x y = 14 3x = y + 14 3x 14 = y x + 2y = 8 2y = -x + 8 2 2 2 y = -1x + 4 2
Graphing in Slope-Intercept Form Y = mx + b b = y-intercept m = slope Why b? Why m? Move to next point . Begin graphing here .
Y = 3x - 4 Move to next point . Begin graphing here .
Equation Forms (review) When working with straight lines, there are often many ways to arrive at an equation or a graph.
There are several ways to graph a straight line given its equation. Let s quickly refresh our memories on equations of straight lines: Slope-intercept y = mx + b When stated in y= form, it quickly gives the slope, m, and where the line crosses the y-axis, b, called the y- intercept. Point-slope y - y1 = m(x x1) when graphing, put this equation into y= form to easily read graphing information. Horizontal line Y = 3 (or any #) Horizontal lines have a slope of zero they have run , but no rise all of the y values are 3. Vertical line X = -2 (or any #) Vertical line have no slope (it does not exist) they have rise , but no run all of the x values are -2.
