Solving a linear system of two variables can be accomplished in four ways: Linear Systems of Two Variables

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1 Hartfield College Algebra (Version Thomas Hartfield) Unit SIX Page of 13 Topic 45: Definition: Linear Systems of Two Variables A system of equations is a set of equations involving the same variables simultaneously. A system of equations is said to be solved when a set of values for each variable in the system satisfies every equation in the system. Systems of linear equations are those systems where every equation is linear. In college algebra we will want to solve systems of linear equations. A system of nonlinear equations is a system with one or more nonlinear equation and will not be covered here. Solving a linear system of two variables can be accomplished in four ways: 1. Finding the intersection point of the graphs of each equation. 2. Substituting one equation into the other to find the value of one variable and then using that value to find the value of the second variable. 3. Adding equations together, after appropriate multiplicative operations to one or both equations, to find the value of one variable. Then using that value to find the value of the second variable. 4. Corresponding the coefficients of the system to a matrix and applying elementary row operations to create an identity matrix and a solution matrix. We will not apply matrices to solve systems in this class. Our primary approaches will be methods 2 and 3, the Substitution Method and the Elimination method, respectively, with supplemental graphing for confirmation.

2 Hartfield College Algebra (Version Thomas Hartfield) Unit SIX Page of 13 The Substitution Method requires the following set of steps: Ex. Solve the system of equations using the Substitution Method. 1. Solve either equation for x or y. 2. Plug that equation in for the variable in the other equation. 3. Solve the created equation involving one variable. 4. Plug the solution found in step 3 into any equation involving two variables & solve. 5. Write your solution as an ordered pair. x 4y 4 3x 2y 16 The Addition/Elimination Method requires the following set of steps: 1. As necessary, multiply one or both coefficients by a positive or negative whole number so that a column of coefficients have the same absolute value but different signs. 2. Add the equations to eliminate a variable and create a new equation. 3. Plug the solution found in step 2 into any equation involving two variables & solve. 4. Write your solution as an ordered pair.

3 Hartfield College Algebra (Version Thomas Hartfield) Unit SIX Page of 13 Ex. Solve the system of equations using the Elimination Method. 2x y 1 3x 2y 4 Observe that if each system is graphed, the solution corresponds with the intersection point of the equations in the system.

4 Hartfield College Algebra (Version Thomas Hartfield) Unit SIX Page of 13 Ex. 3 Solve the system of equations. 5x 3y 2 2x 4y 6 Most systems of linear equations have exactly one solution. A system that has exactly one solution is classified as being consistent and independent. A system of linear equation can have no solution if the graphs of the equations are parallel (as thus lack an intersection point). Such a system is classified as being inconsistent. When solving a system through either method, if a contradiction is reached (such as two different numbers supposedly being equal to each other), the system must be inconsistent. A system of linear equations can have infinitelymany solutions if the equations are not unique (and thus share all their points along a coincident line). Such a system is classified as being dependent. When solving a system through either method, if an identity statement is reached (such as a number being equal to itself), the system must be dependent.

5 Hartfield College Algebra (Version Thomas Hartfield) Unit SIX Page of 13 Topic 46: Linear Systems of Three Variables To solve a linear system of three variables, we will follow the following set of steps: 1. Write each equation in standard form Ax + By + Cz = D 2. Choose a pair of equations and eliminate one of the variables by the Addition/Elimination method. 3. Choose a different pair of equations and eliminate the same variable. 4. Utilize the equations in steps 2 and 3 to solve for one variable by either the Substitution or Addition/Elimination method. Then find the value of the other variable in those equations. 5. Plug the values found in step 4 into any of the original equations to find the value of the third variable. 6. Write your solution as an ordered triple. Observation: If a contradiction or an identity statement is found through the process of rewriting into triangular form, then the system has no solution or infinitely-many solutions, respectively.

6 Hartfield College Algebra (Version Thomas Hartfield) Unit SIX Page of 13 Ex. 1 Solve the system of equations. x y 2z 2 3x y 5z 8 2x y 2z 7

7 Hartfield College Algebra (Version Thomas Hartfield) Unit SIX Page of 13 Ex. 2 Solve the system of equations. x y z0 3x y 6 x 2y 5z 3

8 Hartfield College Algebra (Version Thomas Hartfield) Unit SIX Page of 13 Ex. 3 Solve the system of equations. 2x y 3z 8 x y z 5 2x 4y 5z 1

9 Hartfield College Algebra (Version Thomas Hartfield) Unit SIX Page of 13 Topic 47: Variation In some situations, it is easy to define one or more independent variables that impact a dependent variable. There are three forms of variation scenarios: Direct Variation y kx Inverse Variation y k x Our approach to variation problems will come in three steps: 1. Use the information given to write a general variation equation. 2. Use given data to solve for the constant of proportionality (also called the constant of variation). 3. Merge the general variation equation and the constant of proportionality to make a specific variation equation and solve. Joint Variation y kxz Reminder: All variation equations must include a constant of proportionality k.

10 Hartfield College Algebra (Version Thomas Hartfield) Unit SIX Page of 13 Write a general variation equation from each statement. Write a general variation equation from each statement. Ex. 1 P is directly proportional to u. Ex. 4 S varies directly as the product of the squares of r and. Ex. 2 M varies inversely as t. Ex. 5 A is proportional to the second power of t and inversely proportional to the cube of x. Ex. 3 h is inversely proportional to the product of a and b.

11 Hartfield College Algebra (Version Thomas Hartfield) Unit SIX Page of 13 Ex. 6 Write a general variation equation from each statement and then use the data to find the constant of proportionality. Ex. 7 Write a general variation equation from each statement and then use the data to find the constant of proportionality. t is jointly proportional to x and y and inversely proportional to r. When x = 2, y = 3, and r = 12, t = 25. is proportional to a and inversely proportional to the square of b. When a = 54 and b = 3, = 2.

12 Hartfield College Algebra (Version Thomas Hartfield) Unit SIX Page of 13 Applications of Variation Ex. 8 The intensity of illumination from a light, I, varies inversely as the square of the distance, d, from the light. A particular lamp has an intensity of 1000 candles at 8 yards. What will be the intensity of the lamp at 20 yards? Ex. 9 The pressure of a sample of gas, P, is directly proportional to the temperature, T, and inversely proportional to the volume, V. If 100 L of gas exerts a pressure of 33.2 kpa at 400 K, determine the pressure exerted by the gas if the temperature is raised to 500 K and the volume is reduced to 80 L.

13 Hartfield College Algebra (Version Thomas Hartfield) Unit SIX Page of 13 Ex. 10 The maximum weight, M, a beam can support is jointly proportional to its width, w, and the square of its height, h, and inversely proportional to its length, l. A beam with dimensions as shown in the picture below at left can support 4800 lbs. If a beam made from the same type of wood has the dimensions as shown in the picture below at right, what is the maximum weight it can support?

The Coordinate Plane; Graphs of Equations of Two Variables. A graph of an equation is the set of all points which are solutions to the equation.

Hartfield College Algebra (Version 2015b - Thomas Hartfield) Unit TWO Page 1 of 30 Topic 9: The Coordinate Plane; Graphs of Equations of Two Variables A graph of an equation is the set of all points which