Algebra Tutorials!
Monday 22nd of April
Solving Linear Equations
Systems of Linear Equations
Solving Linear Equations Graphically
Algebra Expressions
Evaluating Expressions and Solving Equations
Fraction rules
Factoring Quadratic Trinomials
Multiplying and Dividing Fractions
Dividing Decimals by Whole Numbers
Adding and Subtracting Radicals
Subtracting Fractions
Factoring Polynomials by Grouping
Slopes of Perpendicular Lines
Linear Equations
Roots - Radicals 1
Graph of a Line
Sum of the Roots of a Quadratic
Writing Linear Equations Using Slope and Point
Factoring Trinomials with Leading Coefficient 1
Writing Linear Equations Using Slope and Point
Simplifying Expressions with Negative Exponents
Solving Equations 3
Solving Quadratic Equations
Parent and Family Graphs
Collecting Like Terms
nth Roots
Power of a Quotient Property of Exponents
Adding and Subtracting Fractions
Solving Linear Systems of Equations by Elimination
The Quadratic Formula
Fractions and Mixed Numbers
Solving Rational Equations
Multiplying Special Binomials
Rounding Numbers
Factoring by Grouping
Polar Form of a Complex Number
Solving Quadratic Equations
Simplifying Complex Fractions
Common Logs
Operations on Signed Numbers
Multiplying Fractions in General
Dividing Polynomials
Higher Degrees and Variable Exponents
Solving Quadratic Inequalities with a Sign Graph
Writing a Rational Expression in Lowest Terms
Solving Quadratic Inequalities with a Sign Graph
Solving Linear Equations
The Square of a Binomial
Properties of Negative Exponents
Inverse Functions
Rotating an Ellipse
Multiplying Numbers
Linear Equations
Solving Equations with One Log Term
Combining Operations
The Ellipse
Straight Lines
Graphing Inequalities in Two Variables
Solving Trigonometric Equations
Adding and Subtracting Fractions
Simple Trinomials as Products of Binomials
Ratios and Proportions
Solving Equations
Multiplying and Dividing Fractions 2
Rational Numbers
Difference of Two Squares
Factoring Polynomials by Grouping
Solving Equations That Contain Rational Expressions
Solving Quadratic Equations
Dividing and Subtracting Rational Expressions
Square Roots and Real Numbers
Order of Operations
Solving Nonlinear Equations by Substitution
The Distance and Midpoint Formulas
Linear Equations
Graphing Using x- and y- Intercepts
Properties of Exponents
Solving Quadratic Equations
Solving One-Step Equations Using Algebra
Relatively Prime Numbers
Solving a Quadratic Inequality with Two Solutions
Operations on Radicals
Factoring a Difference of Two Squares
Straight Lines
Solving Quadratic Equations by Factoring
Graphing Logarithmic Functions
Simplifying Expressions Involving Variables
Adding Integers
Factoring Completely General Quadratic Trinomials
Using Patterns to Multiply Two Binomials
Adding and Subtracting Rational Expressions With Unlike Denominators
Rational Exponents
Horizontal and Vertical Lines
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Graph of a Line

We can graph the linear equation defined by y = x + 1 by finding several ordered pairs. For example, if x = 2 then y = 2 + 1 = 3, giving the ordered pair (2, 3). Also, (0, 1), (4, 5), (-2, -1), (-5, -4), (-3, -2), among many others, are ordered pairs that satisfy the equation.

To graph y = x + 1 we begin by locating the ordered pairs obtained above, as shown in Figure 6(a). All the points of this graph appear to lie on a straight line, as in Figure 6(b). This straight line is the graph of y = x + 1.

It can be shown that every equation of the form ax + by = c has a straight line as its graph. Although just two points are needed to determine a line, it is a good idea to plot a third point as a check. It is often convenient to use the x- and y-intercepts as the two points, as in the following example.


Graph of a Line

Graph 3x + 2y = 12.


To find the y -intercept, let x = 0.

3(0) + 2y = 12  
2y = 12 Divide both sides by 2.
y = 6  

Similarly, find the x-intercept by letting y = 0 which gives x = 4. Verify that when x = 2 the result is y = 3. These three points are plotted in Figure 7(a). A line is drawn through them in Figure 7(b).

Not every line has two distinct intercepts; the graph in the next example does not cross the x-axis, and so it has no x-intercept.


Graph of a Horizontal Line

Graph y = -3.


The equation y = -3 or equivalently y = 0x -3, always gives the same y -value, - 3, for any value of x . Therefore, no value of x will make y = 0, so the graph has no x -intercept. The graph of such an equation is a horizontal line parallel to the x -axis. In this case the y -intercept is - 3, as shown in Figure 8.

In general, the graph of y = k, where k is a real number, is the horizontal line having y-intercept k.

The graph in Example 13 had only one intercept. Another type of linear equation with coinciding intercepts is graphed in Example 14.


Graph of a Line Through the Origin

Graph y = -3x.


Begin by looking for the x -intercept. If y = 0 then

y = -3x  
0 = -3x Let y = 0
0 = x Divide both sides by -3

We have the ordered pair (0, 0). Starting with x = 0 gives exactly the same ordered pair, (0, 0). Two points are needed to determine a straight line, and the intercepts have led to only one point. To get a second point, choose some other value of x (or y ). For example, if x = 2 then

y = -3x = -3(2) = -6, (let x = 2)

giving the ordered pair (2, -6). These two ordered pairs, (0, 0) and (2, -6), were used to get the graph shown in Figure 9.

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