Precalculus by Richard Wright

Are you not my student and
has this helped you?

This book is available

Blessed are those who mourn, for they will be comforted. Matthew‬ ‭5‬:‭4‬ ‭NIV‬‬

8-02 Two-Variable Linear Systems

Summary: In this section, you will:

• Solve two-variable linear systems using elimination.
• Classify types of solutions.

SDA NAD Content Standards (2018): PC.6.1

Solution

Write two equations: one for cost on one for revenue. The costs are $$C = 10,050 + 150x$$ where x is the number of guitars made. The revenue is the amount of money earned, $$R = 300x$$. This question asks for the break even point where cost equals revenue. Replace the C and R with y since they are equal, and write the system of equations.

\left\{\begin{align} y &= 10,050 + 150x \\ y &= 300x \end{align}\right.

Rewrite to make nice columns.

\left\{\begin{align} 150x - y &= -10,050 \\ 300x - y &= 0 \end{align}\right.

Multiply the top equation by −1 and add.

\begin{align} -150x + y &= 10,050 \\ +\quad \underline{300x - y} &= \underline{0} \\ 150x &= 10,050 \end{align}

Solve.

$$x = 67$$

The company would need to sell 67 guitars to break even.

Lesson Summary

Solving Linear Equations by Elimination
1. Write the equations in columns (ax + by = c).
2. Obtain coefficients of one variable that differ only in sign by multiplying the equations by constants.
3. Add the equations and solve the resulting equation. (A variable will be eliminated.)
4. Back-substitute the answer into either original equation and solve.

If all the variables are eliminated and the result is

• True (like 0 = 0), there are infinitely many solutions.
• False (like 0 = 9), there are no solutions.

Practice Exercises

Check to see if the given point is a solution to the system.

1. \left\{\begin{align} x + 6y &= -5 \\ 3x + 2y &= 1 \end{align}\right.; (1, -1)
2. \left\{\begin{align} 2x - y &= \frac{11}{2} \\ 3x + 2y &= 9 \end{align}\right.; $$\left(3, \frac{1}{2}\right)$$
3. Solve the system of equations and classify.

4. \left\{\begin{align} x + 4y &= 0 \\ 3x - y &= 13 \end{align}\right.
5. \left\{\begin{align} 7x - 5y &= -35 \\ 2x + 2y &= 14 \end{align}\right.
6. \left\{\begin{align} 4x + 9y &= -4 \\ -2x - 6y &= 3 \end{align}\right.
7. \left\{\begin{align} 5x - 3y &= 11 \\ -4x + \frac{12}{5}y &= \frac{41}{5} \end{align}\right.
8. \left\{\begin{align} y &= -3x + 2 \\ x &= -\frac{1}{2}y - \frac{1}{2} \end{align}\right.
9. \left\{\begin{align} 17x + 34y &= 2 \\ -51x - 68y &= -9 \end{align}\right.
10. \left\{\begin{align} \frac{2}{3}x - \frac{5}{3}y &= \frac{7}{3} \\ y &= 0.4x - 1.4 \end{align}\right.
11. \left\{\begin{align} -15x + 16y &= 29 \\ 5x - 12y &= -18 \end{align}\right.
12. Problem Solving

13. The Old Testament specified that people had to sacrifice a lamb at the temple for forgiveness of sins, but if they were poor, people could sacrifice a pair of doves. Two groups of travelers went to the temple and needed to purchase their sacrifices. The first group purchased 2 lambs and 3 pairs of doves for a total of $50.70 in today's dollars. The second group purchased 4 lambs and 1 pair of doves for$81.90. The groups tried to find out how much they were charged per lamb and pair of doves.
1. How much were the people charged per lamb and pair of doves?
2. Look up Mark 11:15-19. What did Jesus do when he saw this?
3. What should the temple have been?
14. Sally sells 10 shells at the seashore. A tourist paid her $2 for each perfect shell and$0.50 for each broken shell. If Sally received $11, how many of each type of shell did Sally collect and sell? 15. Jill wants to make 10 L of 20% bleach solution by mixing some 10% solution and some 50% solution. How much of each type of solution should she use? 16. The soccer club has two fee plans. Plan A is a$100 member fee and $5 per game you play. Plan B is no member fee but$15 per game you play. How many games will you have to play for both plans to cost the same and how much will that cost?
17. Johnny invests $500 in two accounts that earn 1% and 0.5% interest. If he earns to$4.25 in interest, how much did he deposit in the accounts?
18. Mixed Review

19. (8-01) Solve by substitution: \left\{\begin{align} x + y = 7 \\ y = x^2 + 1 \end{align}\right.
20. (8-01) Solve by graphing: \left\{\begin{align} x - y = 1 \\ y = \frac{1}{x - 1} \end{align}\right.
21. (7-09) Write a polar equation of an ellipse with $$e = \frac{1}{3}$$ and directrix $$x = -5$$.
22. (7-07) Graph the polar coordinates: $$A\left(4, \frac{2\pi}{3}\right)$$ and $$B\left(-3, \frac{3\pi}{2}\right)$$.
23. (6-05) Evaluate $$\langle 1, -4 \rangle \cdot \langle 6, 3 \rangle$$.

1. Yes
2. No
3. (4, -1)
4. (0, 7)
5. $$\left(\frac{1}{2}, -\frac{2}{3}\right)$$
6. No solution
7. (3, -7)
8. $$\left(\frac{5}{17}, -\frac{3}{34}\right)$$
9. Many solutions
10. $$\left(-\frac{3}{5}, \frac{5}{4}\right)$$
11. Lamb: $19.50, Doves:$3.90; you look it up
12. 4 perfect, 6 broken
13. 2.5 L of 50%, 7.5 L of 10%
14. 10 games, $150 15.$350 at 1%, \$150 at 0.5%
16. (−3, 10), (2, 5)
17. (0, −1), (2, 1)
18. $$r = \frac{5}{3 - \cos \theta}$$
19. −6