Chapter 5: Linear Systems Section 2: Using Algebra
Topics in Algebra 1 © 2001, 2002 Texas Instruments Teacher Notes 5-29
Common Student Errors
•
When they use the substitution method, students may make sign and division errors when
they solve for one of the variables. They may also need to be reminded to use parentheses
when needed. Students might incorrectly distribute expressions.
•
When solving by substitution, students might solve one equation for y (or x) and substitute
back into the same
equation getting a result of 0 = 0, when they should have substituted
back into the other equation.
•
When they use the elimination method, students may miss multiplying every term in an
equation by the appropriate constant.
•
As they continue using the elimination method, many students subtract incorrectly. They
usually subtract the first term correctly, but often forget to subtract the other terms.
Encourage students to choose the multiplier so that they add the equations for the
elimination rather than subtract them.
•
When an algebraic solution results in a statement that is always true, such as 2 = 2 in a
dependent system (coincident lines), or a statement that is never true, such as 2 = 4 in an
inconsistent system (parallel lines), students may be unsure how to state the solution.
• Students often skip checking their solution. In addition to checking their solution by
substituting it back into the original equations, they should also make sure that the solution
is reasonable. For example, in the Student Worksheet problem where Cathy and Brenda are
shopping for tops and jeans, the variable must be positive.
• Students should practice rewriting the system after each step to keep track of their
manipulations.
Encourage students to create good math habits by doing the following.
• Pick the method—graphs, tables, substitution, or elimination—which is best for the system.
First, look at the physical problem, if appropriate, or use geometry and number sense to
analyze the system.
• Remember that for real problems (word problems), some solutions may need to be omitted.
For example, if you need to find the quantity of an item, the solution must be positive.
• Notice when a system obviously has no solution because the lines have the same slope
(i.e., parallel lines), and when a system has an infinite number of solutions because the lines
are the same line.
• Rewrite the system after each step to keep track of the manipulations.
Note: Students who can think through the steps in their mind tend to be impatient with recording each step. However, forming this habit
now will help them in the future, especially when they learn how to solve systems using matrices.
• Write out the mathematics you use to solve the system. Also, write phrases or sentences that
explain your steps. Draw graphs and tables to use as aids.
• Interpret what the solution means in real problems. For example, the recycling problem in
the
Overview subsection requires not only the numeric answer but also an explanation about
what the numbers mean with respect to the price per pound.
Tip: Have students look in newspapers and on the web for graphs and tables of information to show real examples of the importance of
both computations and explaining what the computations mean.
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