4.1 Solving Systems of Linear Equations in Two Variables
Determine If a Value is a Solution
Replace the x and y values in the equations and see if the values satisfies BOTH equations.
Example: Given 3x + y = 1
y = 1 – 2x a) (2,-5)
b) (0,1)
Definitions
Consistent system of equations - system with at least one solution
Dependent system of equations – system with many solutions
Inconsistent system of equations – system with no solutions
Methods to Solve a System of Two Equations and Two Variables
Solving by Graphing
Graph both linear equations
If, they intercept at one point determine the x/y coordinates
If, they don't intersect at one point
Different lines – no solution ( parallel lines )
Same line – infinite number of solutions (
Check your solution
Exercise 1
Solve by graphing:
[20] x - y = 1; 2x + y = 8 [24] 3x + 2y = 8; 6x + 4y = 18
[26] 2x + y = -3; 8x + 4y = -12
Solving by Substitution
1. Isolate (solve for) one of the variables.
2. If both variables are eliminated,
if the equation is true (infinite solutions), if false (no solution), stop
3. Substitute that value in the OTHER equation
4. Solve for the unknown
5. Use either original equation and find other value
6. Check your solution
Exercise 2:
[44]
x = y + 6; x + y = -2
[54]
x - 2y = 8; -2x + 4y = -12
Solve by Elimination (aka Addition/Subtraction)
1. Write the equations in standard form ( Ax + By = C )
2. Clear equations of fractions/decimals (optional)
3. Determine the LCD of coefficients of x and y, choose one
4. Multiply both equations to make both coefficients be additive inverses for the chosen variable.
5. Add the equations [If both variables are eliminated, check to see if the equation is valid (infinite
solutions) or invalid (no solution)]
6. Divide both sides by the coefficient of the remaining variable, solve
7. Substitute this value to get the value of the other variable
8. Check your solution
Exercise 3:
[62]
3x + 2y = 6 ; 5x + 2y = 14
[78]
4x + 2y = 6 ; 6x = 9 - 3y
Hint: If one of the coefficients equals ± 1, it is probably easier to solve by substitution.
4.2 Solving Systems of Linear Equations in Three Variables
Equations and Unknowns
One equation and one unknown can be solved.
Two equations and two unknowns can have a solution.
Three equations and three unknowns can be solved uniquely.
In general, n equations are required to find a solution for n unknowns, except when there is no solution
or an infinite number of solutions.
Determine If a Value is a Solution
Similar to two equations/unknowns, just replace the x, y, and z values in the equations and see if it
satisfies all three equations.
Exercise 1
[08] Does (2,-2,1) satisfy the system
3x + 2y + z = 3 2x - 3y - 2z = 8
-2x + 4y + 3z = -9
Understand the Types of Solutions for 3 variables and 3 Equations
An equation in two variables is a straight line in two dimensions.
An equation in three variables is a plane in three dimensions.
See the illustrations on page 213
Solve a System of Three Equations by Elimination
1. Write all equations in form Ax + By + Cz = D
2. Eliminate one variable from 2 equations using the elimination method
3. Eliminate the same variable using the 3rd equation using the elimination method
4. Use above equations solving the 2X2 system one to solve one of variables
5. Use the 2X2 system to get 2nd variable
6. Plug the 2 values into equation w/3rd variable
7. Check the solution
Exercise 2
Solve:
[14]
x + y + z = 2 ; 3x + y - z = - 2; 2x - 2y + 3z = 15
[20]
3x - 2y + z = 5; 4x - 5y + 2z = 7; 9x - 6y + 3z = 7
[22]
-8x + 4y + 6z =-18; 2x - y - 1.5z = 4.5; 4x - 2y - 3z = 9
4.3 Solving Applications Using Systems of Equations
Solve Application Problems That Translate to a 2X2 System of Equations
1. Select a variable to represent each solution required
2. Write a system of equations
3. Solve the system
Hint: First determine what equation(s) make sense to use and set up a table to fill in.
In Short, Word Problems
Many of these word problems could be solved using only one variable.
Example: If the sum of two numbers is 16 and the difference is 4, you could let one value be x and
the other be 16 – x rather than setting it up as: x + y = 16; x - y = 4 then solving it.
Exercise 1.
Business & Currency:
[14]
Pier 1 sells 2 sizes of pillar candles. The larger sells for $15, the smaller for $10. One day
the number of small candles sold four more than twice the number of larger ones, for a
total of $845. How many of each were sold?
Travel
[26]
Going upstream, it takes 2 hr to travel 36 mi in a boat. Downstream, the same distance
takes 1.5 hr. Find the speed of the boat and the current.
Mixture
[30]
How much of a 45% saline solution and a 30% saline solution must be mixed to produce
20 liters of 39% saline solution?
Review of Chapter 4
<4.1>
12) Determine if (1, 1) is a solution of 2x – y = 7; x + y = 8
18) Solve the system of equations GRAPHICALLY x – y = 4; 2x + 3y = 3
Solve any method
22) 2x + 5y = 8; x – 10y = 9
24) 8x – 2y = 12; y – 4x = 3
28) 1/5 x – 1/3 y = 2; x + y = 2
26) 3x + 2y = 4; 2x – 3y = 7
<4.2>
30) x + y + z = 0; 2x – 4y + 3z = -12; 3x – 3y + 4z = 2
<4.3>
46) Adult tickets cost $7 and children tickets cost $5.50. If 139 tickets sold for $835, how many tickets of each?
52) A plane flying with the wind flies 450 miles in 3 hours. The return trip takes 5 hours. What is aircraft speed
in still air?
53) How much 10% solution is needed to combine with a 60% solution to create 50 ml of a 30% solution?