Category Archives: 1.1 Linear Equations

Rational Equation (no solution)

Example: Solve the rational equation.

{x+4}/{x-3}+1=7/{x-3}

Solution:

{x+4}/{x-3}+1=7/{x-3}

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.

x-3=0

x-3+3=0+3

x=3

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
{x+4}/{x-3}+1=7/{x-3}
Multiply each side of the equation by the least common multiple of the denominators.  For this equation the least common multiple is x-3
(x-3)({x+4}/{x-3}+1)=(x-3)7/{x-3}
Distribute the least common multiple to each term.
(x-3){x+4}/{x-3}+1(x-3)=(x-3)7/{x-3}
Simplify by canceling the common factors.  This should clear any denominators.
x+4+1(x-3)=7
Use the distributive property to simplify.
x+4+x-3=7
Simplify each side of the equation by combining like terms.
2x+1=7
Solve for x by getting x by itself on one side.  Start by subtracting 1 on both sides.
2x+1-1=7-1
2x=6
 Solve for x by getting x by itself on one side.  Next divide both sides by 2.
{2x}/2=6/2
x=3
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.

(x+5)^3-9=x(x+7)(x+8)-6

Solution:

The original equation
(x+5)^3-9=x(x+7)(x+8)-6
Simplify both sides of the equation.  On the left hand side, rewrite the exponent.  On the right hand side, begin to simplify the multiplication.
(x+5)(x+5)(x+5)-9=x(x^2+8x+7x+56)-6
Simplify both sides of the equation.  On the left hand side, begin multiplying.  On the right hand side, combine like terms.
(x+5)(x^2+5x+5x+25)-9=x(x^2+15x+56)-6
Simplify both sides of the equation.  On the left hand side, combine like terms.  On the right hand side use the distributive property.
(x+5)(x^2+10x+25)-9=x^3+15x^2+56x-6
Simplify both sides of the equation.  On the left hand side, continue multiplying.  The right hand side is in simplest form.
x(x^2+10x+25)+5(x^2+10x+25)-9=x^3+15x^2+56x-6
x^3+10x^2+25x+5x^2+50x+125-9=x^3+15x^2+56x-6
Simplify both sides of the equation.  On the left hand side, combine like terms.  The right hand side is in simplest form.
x^3+15x^2+75x+116=x^3+15x^2+56x-6
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 x^3 from each side.  It cancels from each side.
x^3-x^3+15x^2+75x+116=x^3-x^3+15x^2+56x-6
15x^2+75x+116=15x^2+56x-6
Subtract 15x^2 from each side.  It cancels from each side.
15x^2-15x^2+75x+116=15x^2-15x^2+56x-6
75x+116=56x-6
Subtract 56x from each side and simplify. 
75x-56x+116=56x-56x-6
19x+116=-6
Subtract 116 from each side and simplify.
19x+116-116=-6-116
19x=-122
Get x by it self by dividing by 19 on both sides and simplify.
{19x}/19={-122}/19
x={-122}/19