Linear Equations in Two Variables

Introduction Linear equations may have either one or two variables.
An example of a linear equation in one variable is 3x + 2 = 6.
In this equation, the variable is x.

An example of a linear equation in two variables is 3x + 2y = 6.
The two variables are x and y.

Linear equations in one variable will, with rare exceptions, have only one solution.
The solution or solutions can be graphed on a number line.
Linear equations in two variables have infinitely many solutions.

Their solutions must be graphed on the coordinate plane.

Here is how to think about and understand linear equations in two variables.

1.

Memorize the Different Forms of Linear Equations in Two Variables Section Text 1 There are three basic forms of linear equations: standard form, slope-intercept form and point-slope form.
In standard form, equations follow the pattern Ax + By = C.

The two variable terms are together on one side of the equation while the constant term is on the other.
By convention, the constants A and B are integers and not fractions.

The x term is written first and is positive.

Equations in slope-intercept form follow the pattern y = mx + b.

In this form, m represents the slope. The slope tells you how fast the line goes up compared to how fast it goes across. A very steep line has a larger slope than a line that rises more slowly.

If a line slopes upward as it moves from left to right, the slope is positive.
If it slopes downward, the slope is negative.
A horizontal line has a slope of 0 while a vertical line has an undefined slope.

The slope-intercept form is most useful when you want to graph a line and is the form often used in scientific journals.

If you ever take chemistry lab, most of your linear equations will be written in slope-intercept form. Equations in point-slope form follow the pattern y - y1= m(x - x1) Note that in most textbooks, the 1 will be written as a subscript. The point-slope form is the one you will use most often to create equations.
Later, you will usually use algebraic manipulations to transform them into either standard form or slope-intercept form.
2. Find Solutions for Linear Equations in Two Variables by Finding X and Y -- Intercepts Linear equations in two variables can be solved by finding two points that make the equation true.
Those two points will determine a line and all points on that line will be solutions to that equation.

Since a line has infinitely many points, a linear equation in two variables will have infinitely many solutions.
Solve for the x-intercept by replacing y with 0.

In this equation, 3x + 2y = 6 becomes 3x + 2(0) = 6.
3x = 6 Divide both sides by 3: 3x/3 = 6/3 x = 2. The x-intercept is the point (2,0).

Next, solve for the y intercept by replacing x with 0. 3(0) + 2y = 6.

2y = 6 Divide both sides by 2: 2y/2 = 6/2 y = 3.
The y-intercept is the point (0, 3). Notice that the x-intercept has a y-coordinate of 0 and the y-intercept has an x-coordinate of 0.

Graph the two intercepts, the x-intercept (2,0) and the y-intercept (0,3). 2.
Find the Equation of the Line When Given Two Points To find the equation of a line when given two points, begin by finding the slope.
To find the slope, work with two points on the line.
Using the points from the previous example, choose (2,0) and (0,3). Substitute into the slope formula, which is: (y2 -- y1)/(x2 - x1).
Remember that the 1 and 2 are usually written as subscripts.
Using these two points, let x1= 2 and x2 = 0. Similarly, let y1= 0 and y2= 3.

Substituting into the formula gives (3 - 0 )/(0 - 2). This gives - 3/2.
Notice that the slope is negative and the line will move down as it goes from left to right. Once you have determined the slope, substitute the coordinates of either point and the slope - 3/2 into the point slope form. For this example, use the point (2,0).
y - y1 = m(x - x1) = y - 0 = - 3/2 (x - 2) Note that the x1and y1are being replaced with the coordinates of an ordered pair.
The x and y without the subscripts are left as they are and become the two variables of the equation.
Simplify: y - 0 = y and the equation becomes y = - 3/2 (x - 2) Multiply both sides by 2 to clear the fractions: 2y = 2(-3/2) (x - 2) 2y = -3(x - 2) Distribute the - 3.

2y = - 3x + 6.

Add 3x to both sides: 3x + 2y = - 3x + 3x + 6 3x + 2y = 6.
Notice that this is the equation in standard form.
3.

Find the equation of a line when given a slope and y-intercept. Substitute the values of the slope and y-intercept into the form y = mx + b. Suppose you are told that the slope = --4 and the y-intercept = 2.
Any variables without subscripts remain as they are.
Replace m with --4 and b with 2. y = - 4x + 2 The equation can be left in this form or it can be converted to standard form: 4x + y = - 4x + 4x + 2 4x + y = 2 Two-Variable Equations Linear Equations Slope-Intercept Form Point-Slope Form Standard Form