Category Archives: 1.1 Linear Equations

1.1 Linear Equations

Rational Equation

Example: Solve the rational equation.

{2x}/{x^2-7x-8}-7/{x-8}=1/{x+1}

Solution:

{2x}/{x^2-7x-8}-7/{x-8}=1/{x+1}

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-8=0

x-8+8=0+8

x=8

Having x=8 causes a zero in the denominator and the overall expression undefined.  That makes 8 a restricted value .

x+1=0

x+1-1=0-1

x=-1

Having x=-1 causes a zero in the denominator and the overall expression undefined.  That makes -1 a restricted value .

x^2-7x-8=0

(x-8)(x+1)=0

x-8=0 or x+1=0

x=8 or x=-1

This gives the same restrictions we have already accounted for.

With the restriction in mind we will solve the equation.

 

The original equation
 {2x}/{x^2-7x-8}-7/{x-8}=1/{x+1}
The original equation with each denominator factore
 {2x}/{(x-8)(x+1)}-7/{x-8}=1/{x+1}
Multiply each side of the equation by the least common multiple of the denominators.  For this equation the least common multiple is (x-8)(x+1)
{2x}/{(x-8)(x+1)}-7/{x-8}=1/{x+1}
(x-8)(x+1){2x}/{(x-8)(x+1)}-(x-8)(x+1) 7/{x-8}=(x-8)(x+1) 1/{x+1}
Simplify by canceling the common factors.  This should clear any denominators.
(x-8)(x+1){2x}/{(x-8)(x+1)}-(x-8)(x+1) 7/{x-8}=(x-8)(x+1) 1/{x+1}
2x-7(x+1) =x-8
Use the distributive property to simplify.
2x-7(x+1) =x-8
2x-7x-7 =x-8
Simplify each side of the equation by combining like terms.
2x-7x-7 =x-8
-5x-7 =x-8
Solve for x by getting the variables on one side.
-5x-7 =x-8
-5x-x-7 =x-x-8
-6x-7 =-8
Solve for x by getting x by itself on one side.  Start by adding 7 on both sides.
-6x-7 =-8
-6x-7+7 =-8+7
-6x =-1
 Solve for x by getting x by itself on one side.  Next divide both sides by -6.
-6x =-1
{-6x}/-6=-1/-6
x=1/6
Compare your solution to the restricted value.
Since the solution is not the same as the restricted value we can consider it a solution to the original equation

Video Example:

1.1 Linear Equations

Solving a Linear Equation with Faction Coefficients

Example:  Solve the equation.

1 /9 (6t-9)=11/9 t -{t+9}/18

Solution:

Solve the equation by clearing fractions.
1 /9 (6t-9)=11/9 t -{t+9}/18
The least common multiple of the denominators is 18.  Multiply the least common multiple by each term.
1 /9 (6t-9)=11/9 t -{t+9}/18
18*{1 /9} (6t-9)=18*{11/9} t -18*{t+9}/18
As you simplify, the fractions cancel.
18*{1 /9} (6t-9)=18*{11/9} t -18*{t+9}/18
2 (6t-9)=2  (11) t -(t+9)
Use the distributive property and combine like terms.
2 (6t-9)=2  (11) t -(t+9)
12t-18=22 t -t-9
12t-18=21 t -9
Move the variables to one side of the equation and simplify.
12t-18=21 t -9
12t-21t-18=21 t-21t -9
-9t-18=-9
Get the variable by itself on one side of the equation by adding 18 on both sides and dividing by -9.
-9t-18=-9
-9t-18+18=-9+18
-9t=9
{-9t}/-9=9/-9
t=-1

The solution to the equation is -1.

Here is a youtube video that might help as well.

1.1 Linear Equations, Solving 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:

1.1 Linear Equations, Solving Equations

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