Elimination Using Multiplication
Prime Factors
Equations Involving Rational Exponents
Working with Percentages and Proportions
Rational Expressions
Interval Notation and Graphs
Simplifying Complex Fractions
Dividing Whole Numbers with Long Division
Solving Compound Linear Inequalities
Raising a Quotient to a Power
Solving Rational Equations
Solving Inequalities
Adding with Negative Numbers
Quadratic Inequalities
Dividing Monomials
Using the Discriminant in Factoring
Solving Equations by Factoring
Subtracting Polynomials
Cube Root
The Quadratic Formula
Multiply by the Reciprocal
Relating Equations and Graphs for Quadratic Functions
Multiplying a Polynomial by a Monomial
Calculating Percentages
Solving Systems of Equations using Substitution
Comparing Fractions
Solving Equations Containing Rational Expressions
Factoring Polynomials
Negative Rational Exponents
Roots and Radicals
Intercepts Given Ordered Pairs and Lines
Factoring Polynomials
Solving Linear Inequalities
Mixed Expressions and Complex Fractions
Solving Equations by Multiplying or Dividing
The Addition Method
Finding the Equation of an Inverse Function
Solving Compound Linear Inequalities
Multiplying and Dividing With Square Roots
Exponents and Their Properties
Equations as Functions
Factoring Trinomials
Solving Quadratic Equations by Completing the Square
Dividing by Decimals
Lines and Equations
Simplifying Complex Fractions
Graphing Solution Sets for Inequalities
Standard Form for the Equation of a Line
Checking Division with Multiplication
Elimination Using Addition and Subtraction
Complex Fractions
Multiplication Property of Equality
Solving Proportions Using Cross Multiplication
Product and Quotient of Functions
Quadratic Functions
Solving Compound Inequalities
Operating with Complex Numbers
Equivalent Fractions
Changing Improper Fractions to Mixed Numbers
Multiplying by a Monomial
Solving Linear Equations and Inequalities Graphically
Dividing Polynomials by Monomials
Multiplying Cube Roots
Operations with Monomials
Properties of Exponents
Mixed Numbers and Improper Fractions
Equations Quadratic in Form
Simplifying Square Roots That Contain Whole Numbers
Dividing a Polynomial by a Monomial
Writing Numbers in Scientific Notation
Solutions to Linear Equations in Two Variables
Solving Linear Inequalities
Multiplying Two Mixed Numbers with the Same Fraction
Special Fractions
Solving a Quadratic Inequality
Parent and Family Graphs
Solving Equations with a Fractional Exponent
Evaluating Trigonometric Functions
Solving Equations Involving Rational Expressions
Laws of Exponents
Multiplying Polynomials
Vertical Line Test
Solving Inequalities with Fractions and Parentheses
Multiplying Polynomials
Solving Quadratic and Polynomial Equations
Extraneous Solutions
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Intercepts Given Ordered Pairs and Lines

When a line crosses one, or both, of the axes the point of the intersection is labeled “intercept”.

All points on the y-axis have the first element of zero. The second element is a value of y where the line y = b intersects the line x = 0. The point is labeled as (0, b) and is called the y-intercept point.

All points on the x-axis have the second element of zero. The first element is a value of x where the line x = a intersects the line y = 0. The point is labeled as (a, 0) and is called the x-intercept point.

1. For the equation 2x − 3y = 6 complete the two points below.

2x − 3(0) = 6, x = 3 2(0) − 3y = 6, y = - 2

 ( 3 , 0 ) and ( 0 , -2 )

If both the x-intercept and the y-intercept are integers, we can place them in the middle of the table. Then we can find the difference in the x-values for dx and the difference in the y-values for dy and use these differences to build arithmetic sequences for the table.

Let x = 0, then 2( 0 ) − 3y = 6 or y = -2

which gives the point: ( 0 , -2 )

Let y = 0, then 2x − 3( 0 ) = 6 or x = 3

which gives the point: ( 3 , 0 )

Check the dy and dx on your table.

Now you must CHECK the top and bottom points in the table to be sure that they are points on the line:

For the point (- 3, - 4) replace x = - 3 and y = - 4 in the given equation 2x − 3y = 6


For the point (9, 4) replace x = 9 and y = 4 in the given equation 2x − 3y = 6



2. For the equation 2x + 5y = 10 complete the two intercept points below.

2x + 5(0) = 10, x = 5 2(0) + 5y = 10, y = 2

( 5 , 0 ) and ( 0 , 2 )

Plot the points and draw the line through the points.. Put the points on the table below, find the differences dx and dy between the two points. Use these differences to complete the table of arithmetic sequences. Work in the space below:

Check top and bottom points:

2(- 5) + 5y = 10, y = 4

2(10) + 5y = 10, y = -2

Looking at this line from left to right describe how the points “step” or line “slants” up or down?

Sometimes we encounter a problem where we solve Ax + By = C for y to find intercepts and neither one is an integer.

3. Graph 3x – 5y = 7. All of the numbers are prime and have no common factors.

Note: If x = 0 then and if y = 0 then . Points:

Since neither of these is an integer it will be difficult to use the x-intercept or y-intercept to draw the graph. We need to find integers as coordinates for the starting point.

If we solve the literal equation for y we get

Let us examine the equation Now, let us again turn to LOGIC: We know that we will get an integer for y if we find a replacement value for x that will make the sum of the numerators a multiple of 5.

Since - 3 − 7 = -10 we will choose x = -1:

Again, since 12 − 7 = 5 we will choose x = 4:

Now we have starting points with integers: ( -1, -2 ) and (4, -1) . We can plot the points then:


Start at (-1, -2) and note the move up 3 and right 5 to the next point (4, -1)

and repeat the move or down 3 -- left 5 find more points and draw the line

New points ( - 6, - 5 ) ( 9, 4 )

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