At the end of this lesson, students will be able to:
- Solve systems of equations in two variables by substituting for either variable.
- Manipulate standard form equations to isolate a single variable.
- Solve real-world problems using systems of equations.
- Solve mixture problems using systems of equations.
Terms introduced in this lesson:
standard form of a linear equation
Teaching Strategies and Tips
Use Example 2 to motivate the substitution method.
- As the solution consists of fractions, the system is more complicated than any example presented up to this point.
- Emphasize that the graphing method can only provide an approximation. Therefore a different method is needed.
The substitution method:
- An algebraic method; provides exact solutions.
- A technique for replacing an unknown with another expression to obtain a third equation with only one unknown (replacing equals with equals).
- Best used when one of the coefficients of the variables is 1.
Encourage students to isolate the variable with a coefficient of 1 or −1. Students often give themselves extra work by choosing an equation and a variable at random.
- Mixtures do not necessarily pertain to chemistry. See Example 4 and Review Question 7.
- Approach Example 7 with a picture. By the labeling the unknowns in it, the system of equations will be evident.
In Example 2, have students back-substitute x into one of the original equations in case that an error was made in solving for x.
General Tip: In fact, any of the two original equations can be used to find y once x is determined. The easier-looking the equation the better.
Point out in Example 3 that the question can be answered after determining x=117.65. There is no need to back-substitute to find the cost per month.
General Tip: Remind students to write coordinates in correct order, depending on how they labeled the variables in the beginning.
- Students often incorrectly write the value they found first as x.
- For problems where variables other than x and y are used, have students clearly state which variable is independent and which is dependent. See Review Questions 5-9.