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Linear Equations Introduction

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable. A linear equation is special because:

• It has one or two variables.
• No variable in a linear equation is raised to a power greater than 1 or used as the denominator of a fraction.
• When you find pairs of values that make the linear equation true and plot those pairs on a coordinate grid, all of the points for any one equation lie on the same line. Linear equations graph as straight lines.

One Variable:

A linear equation in one unknown x may always be rewritten

ax=b

If a≠0, there is a unique solution

x=b/a

If a = 0, then either the equation does not have any solution, if b ≠ 0 (it is inconsistent), or every number is a solution, if b is also zero.

Two variables:

A common form of a linear equation in the two variables x and y is Y=mx+b

where m and b designate constants (parameters). The constant m determines the slope or gradient of that line and the constant term b determines the point at which the line crosses the y-axis, otherwise known as the y-intercept.

Different forms for 2D equations:

• General (or standard) form
• Slope–intercept form
• Point–slope form
• Two-point form
• Intercept form
• Matrix form
• Parametric form
• 2D vector determinant form

Graph for linear equation:

Let's draw the graph of this equation.

y = 1/2 x+ 2

• Find the x and y values of two points that satisfy the equation
• Plot each point, and then draw a line through the points.

We can start with any two x values we like and then find y for each x by substituting the x values into the equation. Let's start with x = 1.

Let's plot these points and draw a line through them.

Slope: The slope intercept form of a linear equation has the following form where the equation is solved for y in terms of x:

y = a + bx

b is the slope

a is a constant term. It is the y intercept, the place where the line crosses the y axis.

Example

y = -13 + 7x

This equation is in slope intercept form. The y intercept is (0,-13) and the slope is 7.