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Assignment Help with Linear Equations, Graphs and Slope

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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 a0, 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
  • Slopeintercept form
  • Pointslope 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.

Value of x y=1/2 x +2 Value of y
1 y=1/2*1+2=1/2+2 2.5
2 y=1/2*2+2=1+2 3

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

Graph for linear equation

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.

scope
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