Algebra 1 Linear Equations in Two Variables

The rise is found by subtracting y-values

(x1, y1) (x2, y2) rise = y2 - y1

The run is found by subtracting x-values

(x1, y1) (x2, y2) run = x2 - x1

On a graph, the slope measures the steepness of a line

- A line with a positive slope goes up from left to right. - A line with a negative slope goes down from left to right.

Graphing a linear equation with slope and y-intercept

1.) Find the slope and the y-intercept. 2.) Plot the y-intercept on the graph. 3.) Begin at the y-intercept. Move up or down the number of units represented by the numerator of the slope (rise). If the numerator is positive, move up. If the numerator is negative, move down. 4.) From that position, move left or right the number of units represented by the denominator of the slope (run). If the denominator is positive, move right. If the denominator is negative, move left. 5.) Plot this point and draw a line that passes through this point and the y-intercept. This will give you the graph of the line.

To graph an equation using the intercepts:

1.) Find the x-intercept by letting y = 0 and solving for x. 2.) Find the y-intercept by letting x = 0 and solving for y. 3.) Find one additional ordered pair solution so that you have three points with which to plot the line. 4.) Plot the three points on a graph. 5.) Draw a line through the pints. Put an arrow on each end of the line to indicate that the line extends in both directions.

Writing equations in slope-intercept form: Slope = 2 y-intercept = (0, 4/3) y = mx + b y = 2x + 4/3 Slope = -6 y-intercept = (0, 0) y = mx + b y = -6x + 0 y = -6x

1.) Find the y-intercept. 2.) Find the slope m = y2 - y1 / x2 - x1 3.) Substitute the values into the slope-intercept form y = mx + b

To graph a linear equation in two variables:

1.) Find three ordered pairs that are solutions to the linear equation. 2.) Plot the ordered pairs on the graph. 3.) Draw a line through the three points.

Finding the y-intercept of a linear equation:

1.) Let x = 0 2.) Solve for y

Finding the x-intercept of a linear equation:

1.) Let y = 0 2.) Solve for x

Finding the slope of a line that passes through (x1, y1) and (x2, y2)

1.) Put it into the formula m = y2 - y1 / x2 - x1 2.) Write down point-slope form y - y1 = m(x - x1) 3.) Substitute in m, x, and y 4.) Simplify 5.) Solve for y to get slope-intercept form y = mx + b

To find a solution to a linear equation in two variables:

1.) Substitute a value for one of the variables. 2.) Solve the equation to find the value of the other variable.

The point-slope form of an equation of a line whose slope is m and that passes through the point (x1, y1) is given by y - y1 = m (x - x1)

1.) Write the point-slope equation y - y1 = m (x - x1) 2.) Substitute the given values of x, y, and m into the equation 3.) To write the equation in slope-intercept form, solve for y; y = mx + b

We can represent a specific location, or point, in a rectangular coordinate system by using two numbers; an x-value and a y-value.

These two numbers make up an ordered pair; we write an ordered pair using (parentheses).

What if the point given is not the y-intercept?

We can use another form of a linear equation called point-slope form.

Where do you find the value for x?

You simply choose it ;-)

Ordered Pair

is a solution in an equation if it results in a true statement when the values for the x-coordinate and the y-coordinate are substituted into the equation.

Linear Equation in Two Variables

is an equation that can be written in the form Ax + By = C, where A, B, and C are real numbers, but A and B are not both zero. Ax + By = C is called the standard form of a linear equation.

The Rectangular Coordinate System

is made up of a horizontal number line and a vertical number line that intersect to form a right angle. The point at which theses number lines meet is called the origin (0, 0). Each graph is divided into four sections called quadrants.

Origin

is represented by the ordered pair (0, 0), and it has both x- and y-coordinates of 0.

x-Coordinate

is the first number in an ordered pair and it represents the location relative to the x-axis.

y-Coordinate

is the second number in an ordered pair and it represents the location relative to the y-axis.

One way that we can describe the slope of a line on a graph is

m = rise / run

Remember the formula to find slope is

m = y2 - y1 / x2 - x1

Intercept

the point at which a line crosses an axis is called an intercept.

On the x-axis

the positive direction is to the right and the negative direction is to the left.

On the y-axis

the positive direction is up and the negative direction is down.

Slope

the rate of change between two ordered pair solutions to a linear equation is called a slope. The variable m is typically used to represent slope.

Rise

the rise of a line is the change in the vertical position, or the difference between the y-coordinates of two points on the line.

Run

the run of a line is the change in the horizontal position, or the difference between the x-coordinates of two points on the line.

x-Intercept

the x-intercept of a line is the point where the line crosses the x-axis. The x-intercept is an ordered pair with the coordinates (a, 0), where a is a real number.

y-Intercept

the y-intercept of a line is the point where the line crosses the y-axis. The y-intercept is an ordered pair with the coordinates (o, b), where b is a real number.

Making a table of values can help keep the ordered pairs organized.

x + y ---------- -4 | 0 -5 | 1 3 | -7

Quadrants are labeled as follows:

I = RUQ = Right Upper Quadrant II = LUQ = Left Upper Quadrant III = LLQ = Left Lower Quadrant IV = RLQ = Right Lower Quadrant

Graphing vertical lines

If an equation may be written in the form x = a, where a is a real number, then the graph of the equation is a vertical line; that is, it is a straight line that is parallel to the y-axis

Graphing horizontal lines

If an equation may be written in the form y = b, where b is a real number, then the graph of the equation is a horizontal line; that is, it is a straight line that is parallel to the x-axis

When writing an ordered pair, the order is very important.

The x-value always comes first, and the y-value always comes second. the variables in an ordered pair are written in alphabetical order.

Graphing linear equations through the origin

An equation in the form Ax + By = 0, where A and B are not both zero, will pass through the origin and the only intercept will be (0, 0)

NOTE:

Points that lie on an axis are not considered to be in any quadrant.

Most linear equations have one x-intercept and one y-intercept.

Some linear equations only have one intercept.

The slope of a horizontal line is 0.

The slope of a vertical line is undefined. It can also be said to have "no slope."

When solving linear equations in two variables that contains a fraction,

choose multiples of the denominator to end up working with integers.

The slope-intercept form of an equation of a line that has a slope m and a y-intercept (0, b) is given by the formula y = mx + b

in slope-intercept form, the m is only the coefficient of x, the variable part is not included in the slope.