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Fields Medal Prize Winners (1998)




TUTORIALS:


Solving Quadratic Equations by Using the Quadratic Formula
Addition with Negative Numbers
Solving Linear Systems of Equations by Elimination
Rational Exponents
Solving Quadratic Inequalities
Systems of Equations That Have No Solution or Infinitely Many Solutions
Dividing Polynomials by Monomials and Binomials
Polar Representation of Complex Numbers
Solving Equations with Fractions
Quadratic Expressions Completing Squares
Graphing Linear Inequalities
Square Roots of Negative Complex Numbers
Simplifying Square Roots
The Equation of a Circle
Fractional Exponents
Finding the Least Common Denominator
Simplifying Square Roots That Contain Whole Numbers
Solving Quadratic Equations by Completing the Square
Graphing Exponential Functions
Decimals and Fractions
Adding and Subtracting Fractions
Adding and Subtracting Rational Expressions with Unlike Denominators
Quadratic Equations with Imaginary Solutions
Graphing Solutions of Inequalities
FOIL Multiplying Polynomials
Multiplying and Dividing Monomials
Order and Inequalities
Exponents and Polynomials
Fractions
Variables and Expressions
Multiplying by 14443
Dividing Rational Expressions
Division Property of Radicals
Equations of a Line - Point-Slope Form
Rationalizing the Denominator
Imaginary Solutions to Equations
Multiplying Polynomials
Multiplying Monomials
Adding Fractions
Rationalizing the Denominator
Rational Expressions
Ratios and Proportions
Rationalizing the Denominator
Like Radical Terms
Adding and Subtracting Rational Expressions With Different Denominators
Percents and Fractions
Reducing Fractions to Lowest Terms
Subtracting Mixed Numbers with Renaming
Simplifying Square Roots That Contain Variables
Factors and Prime Numbers
Rules for Integral Exponents
Multiplying Monomials
Graphing an Inverse Function
Factoring Quadratic Expressions
Solving Quadratic Inequalities
Factoring Polynomials
Multiplying Radicals
Simplifying Fractions 1
Graphing Compound Inequalities
Rationalizing the Denominator
Simplifying Products and Quotients Involving Square Roots
Standard Form of a Line
Multiplication by 572
Adding and Subtracting Fractions
Multiplying Polynomials
Factoring Trinomials
Solving Exponential Equations
Solving Equations with Fractions
Roots
Simplifying Complex Fractions
Multiplying and Dividing Fractions
Mathematical Terms
Solving Quadratic Equations by Factoring
Factoring General Polynomials
Adding Rational Expressions with the Same Denominator
The Trigonometric Functions
Solving Nonlinear Equations by Factoring
Solving Systems of Equations
Midpoint of a Line Segment
Complex Numbers
Graphing Systems of Equations
Reducing Rational Expressions
Powers
Rewriting Algebraic Fractions
Exponents
Rationalizing the Denominator
Adding, Subtracting and Multiplying Polynomials
Radical Notation
Solving Radical Equations
Positive Integral Divisors
Solving Rational Equations
Rational Exponents
Mathematical Terms
Rationalizing the Denominator
Subtracting Rational Expressions with the Same Denominator
Axis of Symmetry and Vertex of a Parabola
Simple Partial Fractions
Simplifying Radicals
Powers of Complex Numbers
Fields Medal Prize Winners (1998)

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Equations of a Line

An equation in two first-degree variables, such as has a line as its graph, so it is called a linear equation. In the rest of this section, we consider various forms of the equation of a line. 4 x + 7 y = 20, has a line as its graph, so it is called a linear equation. In the rest of this section, we consider various forms of the equation of a line.

The slope-intercept form of the equation of a line involves the slope and the y-intercept. Sometimes, however, the slope of a line is known, together with one point (perhaps not the y-intercept) that the line goes through. The point-slope form of the equation of a line is used to find the equation in this case. Let (x1, y1) be any fixed point on the line and let (x, y) represent any other point on the line. If m is the slope of the line, then by the definition of slope,

or

y - y1 = m(x - x1)

 

Point-slope form

If a line has slope m and passes through the point (x1, y1), then an equation of the line is given by

y - y1 = m(x - x1)

the point-slope form of the equation of a line.

 

Example

Point-Slope Form

Find an equation of the line that passes through the point (3, -7) and has slope m = 5/4.

Solution

Use the point-slope form.

The equation of the same line can be given in many forms. To avoid confusion, the linear equations used in the rest of this section will be written in slopeintercept form, y = mx + b, which is often the most useful form.

The point-slope form also can be useful to find an equation of a line if we know two different points that the line goes through. The procedure for doing this is shown in the next example.

Example

Using Point-Slope Form to Find Equation

Find an equation of the line through (5, 4) and (-10, -2).

Solution

Begin by using the definition of slope to find the slope of the line that passes through the given points.

Either (5, 4) or (-10, -2) can be used in the point-slope form with m = 2/5. If (x1, y1) = (5, 4) then

y - y1 = m(x - x1)

Check that the same result is found if (x1, y1) = (-10, -2).

Example

Horizontal Line

Find an equation of the line through (8, -4) and (-2, -4).

Solution

Find the slope.

Choose, say, (8, -4) as (x1, y1).

y - y1 = m(x - x1)  
y - (-4) = 0(x - 8) Let y1 = -4, m = 0, x1 = 8
y + 4 = 0 0(x - 8) = 0
y = -4  

Plotting the given ordered pairs and drawing a line through the points, show that the equation y = -4 represents a horizontal line. See Figure 5(a). Every horizontal line has a slope of zero and an equation of the form y = k where k is the y-value of all ordered pairs on the line.

Example

Vertical Line

Find an equation of the line through (4, 3) and (4, -6).

Solution

The slope of the line is

which is undefined. Since both ordered pairs have x-coordinate 4, the equation is x = 4. Because the slope is undefined, the equation of this line cannot be written in the slope-intercept form.

Again, plotting the given ordered pairs and drawing a line through them show that the graph of x = 4 is a vertical line. See Figure 5(b).

The slope of a horizontal line is 0.

The slope of a vertical line is undefined.