Linear Equations and Graphs

Study Notes

Linear Equations and Graphs

Linear equations are mathematical expressions that describe straight-line relationships between two variables. These equations are of the first degree because they only involve the first power of their variable terms. In linear functions, there is one independent variable, and it is a constant with respect to any other variable. In general, linear equations can be written as follows:

ax + by = c

where x and y are variables, a, b, and c are constants. These equations can represent various real-life situations, such as finding the equation of a line passing through two given points or determining how much money you save when purchasing items at different prices.

Finding x-intercepts

To find the x-intercept of a linear equation, substitute y = 0 into the equation and solve for x:

Find the term without y.
Change all terms containing y into zero.
Solve for x.

For example, let's find the x-intercept of the equation 2x + 4y = 8:

2x + 4(0) = 8 2x = 8 x = 4

So, the x-intercept of the equation 2x + 4y = 8 is 4.

Finding y-intercepts

The y-intercept represents the point where the graph of the linear function intersects the y-axis. To find the y-intercept, set x = 0 in the equation and solve for y:

Isolate the term with y.
Set x equal to 0.
Solve for y.

Continuing from the previous example, to find the y-intercept of the equation 2x + 4y = 8, we have:

2(0) + 4y = 8 4y = 8 y = 2

So, the y-intercept of the equation 2x + 4y = 8 is 2.

Graphing lines

Graphing linear functions involves plotting points that satisfy the equation and connecting them. For example, consider the equation y = -2x + 3. To graph this equation:

Evaluate the equation at different values of x (e.g., x = 0, x = 1, x = 2).
Plot these points on a coordinate plane.
Connect the points using a straight line.

The resulting graph will be a line with slope -2 and y-intercept 3.

Slope-intercept form

Slope-intercept form represents a linear equation as y = mx + b, where m is the slope and b is the y-intercept. The slope is a measure of how steep the line is, while the y-intercept tells us where the line intersects the y-axis.

For example, the equation y = 3x + 2 is in slope-intercept form. The slope is 3, which indicates the line rises 3 units for every unit it moves to the right. The y-intercept is 2, which tells us the line crosses the y-axis at a point with coordinates (0, 2).

Point-slope form

Point-slope form represents a linear equation as Ax + By = C, where (x_1, y_1) and (x_2, y_2) are the points on the line. This form is useful when finding the equation of a line that passes through two given points.

For example, consider the points (1, 3) and (2, 4). To find the equation of the line passing through these points using point-slope form, we have:

(y_2 - y_1) / (x_2 - x_1) * (x - x_1) + y_1 = y

Substituting the values:

(4 - 3) / (2 - 1) * (x - 1) + 3 = y

Simplifying:

x - 1 = y

So, the equation of the line passing through the points (1, 3) and (2, 4) is x - 1 = y.

Description

Explore linear equations, x-intercepts, y-intercepts, graphing lines, slope-intercept form, and point-slope form in this quiz. Learn how to find x-intercepts, y-intercepts, graph linear functions, interpret slope in equations, and derive equations of lines passing through given points.