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DrDelMath

College Algebra Examples
Equations: Solution and Solve


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The definition that pops up when you move the cursor over a highlighted word is what should pop into your mind when you read, hear, or speak that word. The material that pops up when you move the cursor over a light bulb is a suggested strategy for solving the problem. You should always formulate a similar strategy when you attempt to solve a problem. When you move the cursor over an explanation of a step in a solution, the property that pops up is the mathematical justification for taking that action. You must always be able to provide such a justification for every step you take in mathematics.

1. Is 5 a solution of 3x + 2 = 8 ?
 If x is replaced with 5 the equation becomes 15 + 2 = 8 which is FALSE. Therefore 5 is not a solution of the equation.

2. Is 2/3 a solution of 3x + 2 = 0 ?
 If x is replaced with 2/3 the equation becomes 2 + 2 = 0 which is FALSE. Therefore 2/3 fails to satisfy the definition of solution. Hence 2/3 is not a solution of the equation.

3. Is 2/(-3) a solution of 3x + 2 = 0 ?
 If x is replaced with 2/(-3), the equation becomes -2 + 2 = 0 which is TRUE. Therefore 2/(-3) satisfies the definition of solution. Hence 2/(-3) is a solution of the equation.

4. Is -4 a solution of x2 = 16 ?
 If x is replaced with -4, the equation becomes 16 = 16 which is TRUE. Therefore -4 satisfies the definition of solution. Hence -4 is a solution of the equation.

5. Is {-4} the solution set for x2 = 16 ?
 From Example 4, we know that -4 is a solution of the equation. That means -4 is in the solution set for the equation. However, 4 is also a solution of the equation and 4 is not in the set {-4}. Therfore {-4} is not the solution set for the equation.

6. What is the solution set for x = 7 ?
 It is pretty obvious that the only number which makes the equation a true statement is 7. Therefore the solution set for the equation is {7}. Notice this is a simplest equation. One of the VERY useful properties of a simplest equation is the fact that its solution set is obvious.

7. What is the solution set for x = 2/3 ?
 It is pretty obvious that the only number which makes the equation a true statement is 2/3. Therefore the solution set for the equation is {2/3}. Notice this is a simplest equation. One of the VERY useful properties of a simplest equation is the fact that its solution set is obvious.

8. What is the solution set for x = ?
 It is pretty obvious that the only number which makes the equation a true statement is . Therefore the solution set for the equation is {}. Notice this is a simplest equation. One of the VERY useful properties of a simplest equation is the fact that its solution set is obvious.

9. Is 3x5 + 2 = 5 equivalent to 3x5 + 2 + (5x3 - 7) = 5 + (5x3 - 7) ?
 The second equation is obtained from the first equation by adding the expression (5x3 - 7) to both sides of the first equation. The first property of equations states that if the same expression is added to both sides of an equation, the resulting equation is equivalent to the original equation. Therefore these two equations are equivalent.

10. Is 3x + 7 = 9 equivalent to 3x = 2?
 The second equation is obtained from the first equation by adding the expression - 7 to both sides of the first equation. The first property of equations states that if the same expression is added to both sides of an equation, the resulting equation is equivalent to the original equation. Therefore these two equations are equivalent.

11. Is 3x = 2 equivalent to x = 2/3 ?
The second equation is obtained from the first equation by multiplying both sides of the first equation by 1/3. The second property of equations states that if both sides of an equation are multiplied by a non-zero real number, the resulting equation is equivalent to the original equation. Therefore these two equations are equivalent.

12. Suppose both sides of the equation 3x3 + 7x - 2 = 5 are multiplied by 2x to obtain the second equation 6x4 + 7x2 - 2x = 5x. Are these two equations equivalent ?
Only the second property of equations deals with multiplication and it is very clear that the conclusion follows only if both sides of an equation are multiplied by a non-zero real number. Therefore we may NEVER conclude that multiplication by an expression containing a variable will produce an equivalent equation.
If both sides of an equation are multiplied by an expression containing a variable, it must be assumed that the resulting equation is not equivalent to the original equaion.

OBSERVE: According to Example 7, the solution set for x = 2/3 is {2/3} and according to Example 11, {2/3} is the solution set for 3x = 2. Then according to Example 10, {2/3} is the solution set for 3x + 7 = 9. This is the logic and the process used to solve many equation.

Remark: The two properties of equations are all that is needed to solve a linear equation, they form the basis for solving many other kinds of equations, and they are also used in a variety of instances which do not involve solving equations.