# Solving an Exponential Equation: Relating the Bases

Example:  Solve the exponential equation.

Solution:

 The exponential equation Try to write both sides of the equation with the same base.  Try 4 since there is a base of 4 on the left Using a property of negative exponents move the base to the numerator Now that that the bases are the same the exponents must be equal Solve for x

The solution the the exponential equation is 4.

# Rational Equation (no solution)

Example: Solve the rational equation.

Solution:

Since we are solving a rational equation we need to first find the restrictions (the values of x that cause the expression to be undefined).

To find the restrictions create an equation by setting each denominator equal to zero and solving.

Having x=3 causes a zero in the denominator and the overall expression undefined.  That makes 3 a restricted value .

With the restriction in mind we will solve the equation.

 The original equation Multiply each side of the equation by the least common multiple of the denominators.  For this equation the least common multiple is Distribute the least common multiple to each term. Simplify by canceling the common factors.  This should clear any denominators. Use the distributive property to simplify. Simplify each side of the equation by combining like terms. Solve for x by getting x by itself on one side.  Start by subtracting 1 on both sides. Solve for x by getting x by itself on one side.  Next divide both sides by 2. Compare your solution to the restricted value. Since the solution is the same as the restricted value we must exclude it as a solution.  Since all of the solutions have been excluded, there is no solution to the rational equation.

Video Example:

# Higher Order Equation that reduces to a linear equation

Example: Solve the equation.

Solution:

 The original equation Simplify both sides of the equation.  On the left hand side, rewrite the exponent.  On the right hand side, begin to simplify the multiplication. Simplify both sides of the equation.  On the left hand side, begin multiplying.  On the right hand side, combine like terms. Simplify both sides of the equation.  On the left hand side, combine like terms.  On the right hand side use the distributive property. Simplify both sides of the equation.  On the left hand side, continue multiplying.  The right hand side is in simplest form. Simplify both sides of the equation.  On the left hand side, combine like terms.  The right hand side is in simplest form. Now that each side is in simplest form we want the terms with x on one side and the constant terms on the the other side.  Subtract from each side.  It cancels from each side. Subtract from each side.  It cancels from each side. Subtract from each side and simplify. Subtract from each side and simplify. Get x by it self by dividing by 19 on both sides and simplify.

# Forms of Linear Equations

Slope Intercept Form

m is the slope of the line and is the y-intercept

Point Slope Form

m is the slope of the line and is a point on the line.

Standard Form of a Line

# Perpendicular Lines and Parallel Lines

Example: What are parallel and perpendicular lines?

Example: How are the slopes of parallel and perpendicular lines related? (only watch until 1 min 20 seconds)

Example: Are the lines parallel, perpendicular or neither?

Example: Are the lines perpendicular to each other?

# Finding the equation of a line perpendicular to another line

Example: Find the equation of a line perpendicular to another line and passing through a specific point. (The other line in slope intercept form)

Example: Find the equation of a line perpendicular to another line and passing through a specific point. (The other line in slope standard form)

Example: Find the equation of a line perpendicular to the x-axis.

Example: Find the equation of a line perpendicular to the x-axis and perpendicular to the y-axis.

# Finding the Equation of a Line parallel to another line

Example: Find the equation of the line parallel to another line and passing through a specific point. (parallel equation in slope intercept form)

Example: Find the equation of the line parallel to another line and passing though a specific point. (parallel line in standard form)

Example: Find the equation of the line parallel to the x-axis or y-axis and passing through a specific point.

Example: What is an equation parallel to the y-axis?

Example: What is an equation parallel to the x-axis?

# Finding the Equation of a line given a fractional slope and a point

Example:  Find the equation of a line in slope intercept form given the slope of the line is and the line passes through the point

Solution:

Use the point-slope formula of the line to start building the line.  m represents the slope of the line and is a point on the line.

Point-slope formula:

and

Substitute the values into the formula.

Since the instructions ask to write the equation in slope intercept form we will simplify and write the equation with y by itself on one side.  I will also use the clearing fractions method to avoid having to add fractions.

(Multiply by LCM)

(Cancel Denominator)

The equation of a line in slope intercept form with a slope of  and  passing through the point  is

# Finding the Equation of a Line given two points on the line

Example:  Find the equation of a line in slope intercept form given the line passes through the two points and .

Solution:

First find the slope of the line.

Choose one of the points to be and choose the other point to be .

I will choose   to be   and choose to be .

Substitute these values into the slope formula and simplify.

The slope of the line containing the points and   is .

Then, use the point-slope formula of the line to start building the line.  m represents the slope of the line and you can use or as the point on the line.

Point-slope formula:

and

Substitute the values into the formula.

Since the instructions ask to write the equation in slope intercept form we will simplify and write the equation with y by itself on one side.

The equation of a line in slope intercept form passing through the two points and is  .