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 Friday 20th of September

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 Depdendent Variable

 Number of equations to solve: 23456789
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 Dependent Variable

 Number of inequalities to solve: 23456789
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 Solve for:

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# Solving Linear Equations

The two examples below show that it is not true that the solution to every linear equation is a unique real number.

Example 1

Solve: 4(x - 2) = 10x + 5 - 6x

 Solution 4(x - 2) = 10x + 5 - 6x Step 1 Remove parentheses. Distribute the 4. 4x - 8 = 10x + 5 - 6x Step 2 On each side of the equation, combine like terms. Combine 10x and -6x. 4x - 8 = 4x + 5 Step 3 Isolate the variable. Subtract 4x from both sides. Simplify. 4x - 8 - 4x -8 = 4x + 5 - 4x = 5

The result, -8 = 5 is a not a true statement. So, there is no value of x which will make the original equation true.

Therefore, the equation 4(x - 2) = 10x + 5 - 6x has no solution.

Note:

The statement -8 = 5 is a false statement, which is sometimes called a contradiction.

Example 2

Solve: 3(y - 2) + 2y = -6 + 5y

 Solution 3(y - 2) + 2y = -6 + 5y Step 1 Remove parentheses. Distribute the 3. 3y - 6 + 2y = -6 + 5y Step 2 On each side of the equation, combine like terms. Combine 3y and 2y. 5y - 6 = -6 + 5y Step 3 Isolate the variable. Subtract 5y from both sides. Simplify. 5y - 6 - 5y -6 = -6 + 5y - 5y -6

The result, -6 = -6 is a true statement. Therefore, any value of y will make the original equation true.

So, the solution of 3(y - 2) + 2y = -6 + 5y is all real numbers.

Note:

Here is another way to solve 5y - 6 = -6 + 5y.

 Add 6. Divide by 5. 5y y = 5y = y

The statement y = y is true for all real numbers.

The statement - 6 = -6 is a true statement, which is sometimes called an identity.