Static Equilibrium Analysis: An Algebraic Approach

J. Zachary Klingensmith

2 Static Equilibrium Analysis: An Algebraic Approach

2.1 Equilibrium

Practice Problems

There are no practice problems for this section.

External Resources

There are no external resources for this section.

2.2 Using algebra to solve Linear systems: Two equations

Practice Problems

Problem 2.2.1: Solve the following system of equations, if possible, with whatever method you would like (other than a calculator.)

$4x+2y=-10$
$3x+9y=0$

Solutions: $(x,y)=(-3,1)$

Problem 2.2.2: Solve the following system of equations, if possible, with whatever method you would like (other than a calculator.)

$-x+2y=-1$
$5x-10y=6$

Solution: Inconsistent System (No Solutions)

Problem 2.2.3: Solve the following system of equations, if possible, with whatever method you would like (other than a calculator.)

$0.1x+0.2y=2$
$0.35x-0.3y=0$

Solution: $(x,y)=(6,7)$

Problem 2.2.4: Solve the following system of equations, if possible, with whatever method you would like (other than a calculator.)

$2x-y=5$
$10x-5y=1$

Solution: Inconsistent System (No Solution)

Problem 2.2.5: Solve the following system of equations, if possible, with whatever method you would like (other than a calculator.)

$3x+6y=12$
$2x+4y=8$

Solution: Dependent solution (infinite number of solutions)

Problem 2.2.6: A moving company charges a flat rate of 150USD,and an additional 5USD for each box. If a taxi service would charge 20USD for each box, how many boxes would you need for it to be cheaper to use the moving company, and what would be the total cost?

Solution: Use moving company for more than 10 boxes.

Problem 2.2.7: If an investor invests 23,000USD into two bonds, one that pays 4% in simple interest, and the other paying 2% simple interest, and the investor earns 710.00USD annual interest, how much was invested in each account?

Solution: 12,500 deposited in first bond, 10,500 deposited in second bond

External Resources

Khan Academy: System of Equations using Elimination

Khan Academy: System of Equations using Substitution

Khan Academy: The Number of Solutions to a System of Equations

2.3 Using algebra to solve linear systems: Three or more variables

Practice Problems

Problem 2.3.1: Solve the following system of equations, if possible, with whatever method you would like (other than a calculator.)

$x-2y+3z=7$
$2x+y+z=4$
$-3x+2y-2z=-10$

Solution: $(x,y,z)=(2,-1,1)$

Problem 2.3.2: Solve the following system of equations, if possible, with whatever method you would like (other than a calculator.)

$3x-4y+2z=-15$
$2x+4y+z=16$
$2x+3y+5z=20$

Solution: $(x,y,z)=(-1,4,2)$

Problem 2.3.3: Solve the following system of equations, if possible, with whatever method you would like (other than a calculator.)

$2x+3y-6z=1$
$-4x-6y+12z=-2$
$x+2y+5z=10$

Solution: Dependent system (no solution)

Problem 2.3.4: At a carnival, 10,400USD in receipts were taken at the end of the day. The cost of a child’s ticket was 20USD, an adult ticket was 30USD, and a senior citizen ticket was 15USD. There were twice as many senior citizens as adults in attendance, and 20 more children than senior citizens. How many children, adult, and senior citizen tickets were sold?

Solution: 220 children, 100 adults, 200 seniors

Problem 2.3.5: You inherit one million dollars. You invest it all in three accounts for one year. The first account pays 3% compounded annually, the second account pays 4% compounded annually, and the third account pays 2% compounded annually. After one year, you earn 34,000USD in interest. If you invest four times the money into the account that pays 3% compared to 2%, how much did you invest in each account?

Solution: 400,000 in account 1 (3%), 500,000 in account 2 (4%), 100,000 in account 3 (2%)