Overview:
Solving equations in algebra depends on using equivalent sentences until the solution is found. Sentences are equivalent if they have the same solution. For example, sentences such as x +3 = 8 and 2x = 10 are equivalent because in both of them x = 5.
How to Use Equivalent Sentences
When solving equations, each sentence is equivalent. Suppose you have the equation x +3 = 8. In the next step, subtract -3 from both sides of the sentence to have the equivalent sentence x +3-3 = 8-3. That means that x = 8-3, or x = 5. The solution to the problem, x=5, is equivalent to the original sentence, x +3 = 8.
Checking for Errors
By using equivalent sentences, students can often pinpoint exactly where they went wrong in attempts to solve an equation. For example, suppose the original equation was -6 = y-9. In the next step, -6 + 9 = y-9 + 9, which gives the equivalent sentence -6 + 9 = y. The correct answer is 3 = y. It checks in the original equation, as -6 = 3-9. Suppose that in the second step, a student subtracted the 9 instead of adding the 9 to the -6, to get the solution y =-15. The -15 is not correct in the original equation, as -6 is not equal to ( ≠ ) -15 – 9.
When Sentences Are Not Equivalent
Very often, it is a matter of knowing what sentence cannot be an equivalent solution, because it is not equivalent to the others. In the previous paragraph, x = -15 is not equivalent to the original sentence, -6 = y – 9. Similarly, if x + 3 = 8 and 2x = 10, the equation x -0.4 = 4.96 does not have an equivalent solution, because x -0.4 +0.4 = 4.96 + 0.4 or x = 5.36. The variable x is not equivalent in all three equations if it equals 5 in two of them and 5.36 in one of them.
Equivalent Formulas
Students often use formulas to find a solution to an application problem in math. For example, the formula for the area of a rectangle is A = lw, where l is the length and w is the width. Sometimes, one might need to solve the equation for one of the variables instead. When solving for w, the equivalent formula is A/l = w, and when solving for l the equivalent formula is A/w = l.
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