Solving Systems of Linear Equations


 Domenic Thornton
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1 Solving Systems of Linear Equations What is a system of equations? A set of equations, for example, two equations with two unknowns, for which a common solution is sought is called a system of equations. We will cover 4 ways that we can use to solve a system of linear equations. Graphically Graph both equations and find the intersection point. Can be inaccurate when done by hand. Useful when using technology. Solve for y first so equation is in the form y=mx+b, then plot. Substitution Solve one equation for one variable and then substitute that into the other equation. Works well when a variable can be solved for easily, has a coefficient of one. Works better when fractions and roots aren t involved. Elimination Multiply one or more equations by a constant to create opposite coefficients and then add the two equations together to eliminate one variable. Works well for a linear system when there is no variable with a coefficient of one. Best to use when both equations have coefficients with both variables (i.e. both x and y have numbers with them). Matrices Uses the inverse of a matrix to find the solution. Works only for square systems of linear equations where the determinant of the coefficient matrix isn t zero. Good for a computer or calculator where there is a matrix inverse function. Quick on most graphing calculators. Graphing Method The graphical approach works well with a graphing calculator, but may be inaccurate by hand (did those points intersect at 1/6 or 1/7?) unless the graph happens to fall exactly on the grid lines. STEP #1: Graph the first equation on the Cartesian plane using an appropriate method... Table of Values SlopeIntercept: use y=mx+b ; graph the yintercept (where x = 0) and use the slope to find other points Intercept method STEP #2: Graph the other equations on the Cartesian plane. STEP #3: Find the point(s) that the two lines have in common (points of intersection). STEP #4: Interpret the results and write a sentence.
2 Example: A CHILD S RIDDLE Can you think of two numbers that when added together total 7, but when subtracted from each other their difference is 1? SOLUTION: 4 and 3 Graph the system: x+y=7 xy = 1 1. You should notice that the x and y values along the dark line satisfy the first condition. That is, the sum of two numbers is Notice the x and y values along the light line satisfy the second condition. Namely, the difference of two numbers is Since the point (4,3) is the intersection of both lines it satisfies both conditions and is called the solution to the system. Extra Hints for Graphing: When you graph the lines, there are three possible outcomes... APPEARANCE MEANING RESPONSE Parallel lines? no points in common no solution Intersecting lines? one point in common that point IS the solution Same line? all points in common infinitely many solutions How do you know what the window settings should be? Good question. There is no simple rule. First, consider what is reasonable for the scenario you are given (If x = amount of exercise per day, a reasonable xaxis would go from 0 to about 6 hours. If they run marathons, maybe 10 to 12 hours). Also, consider your yintercept. This can help you
3 know at least where the graph is going to start. Sometimes I do a quick rough sketch on scrap paper first. Then view the graph and adjust as necessary. Strange Slope: If the slope is a decimal, sometimes it is useful to convert it to an equivalent fraction with whole numbers. Substitution It is important that both answers be given when solving a system of equations. A common mistake students make is to find one variable and stop there. You need to include a value for all the variables. STEP #1: Write equations based on the problem. Solve one of the equations for one of the variables. STEP #2: Substitute that expression in for the variable in the other equation. STEP #3: Solve the equation for the remaining variable STEP #4: Backsubstitute the value for the variable to find the other variable. The process of backsubstitution involves taking the value of the variable found in step 3 and substituting it back into the expression obtained in step 1 (or the original problem) to find the remaining variable. STEP #5: Interpret your answer and write a sentence. Example 1: Which is the better value when renting a vehicle? RentAHunk o Junk charges $29.95 per day and 43 per mile. Tom s Total Wrecks charges $45 per day plus 32 per mile. If you knew when the costs were the same, you could easily determine the better value for your individual needs. STEP 1: First, let s choose some meaningful variables to describe our situation. Let m = the total miles to be driven. Let c = the total rental cost for each company.
4 Then write a system of two equations using the two unknown variables. c= m Cost for RentAHunk o Junk c= m Cost for Tom s Total Wrecks STEP 2: Since both equations are names for cost, c, we can set them equal to each other. This will let us solve for miles, m. STEP 4: Substitute m back into one equation to determine the cost: C = m C = (136.81) c = $ STEP 5: Interpret results and write a sentence. The cost is the same ($88.78) for both rental companies if you plan to drive 137 miles. For distances under 137 miles Rent A Hunk o Junk is cheaper. For distances over 137 miles Tom s Total Wrecks is cheaper.
5 Example 2: Example 2a + 3b = 2 a 2b = 8 a  2b = 8 a = 2b + 8 2a + 3b = 2 2( 2b + 8) +3b = 2 4b b = 2 7b + 16 = 2 7b = b = 14 b=2 a = 2b + 8 a = 2(2) + 8 a = a = 4 The system s solution is at the coordinates (4, 2). Extra Hints for The Substitution Method Method Write equations based on the problem. (This is done for us) Solve one of the equations for one of the variables. (Solve for a ) Substitute the new expression for variable a into the other equation. Solve for b. Back Substitute the value for b into the second equation, to determine the value for a. Interpret your answer and write a sentence. When should you use substitution? Consider these scenarios: This system is easily solved by substitution because one equation has the yterm isolated ready to be substituted into the other equation. This system could be solved by substitution by first solving the second equation for a, then substituting that expression back into the first equation for a. This system is similar to the realworld application at the beginning. Both equations represent a name for m. Therefore, simply set the equations equal to each other and solve for n.
6 Elimination Another method for solving systems of equations is called the elimination or the addition method. It is most useful when both equations are written in standard form Ax + By = C. STEP #1: Write equations based on the problem. Align the components of the system. (Make sure all the variables are lined up, and the = are lined up.) STEP #2: Look for patterns (i.e. Least Common Multiple) to make one of the coefficients the opposite of the other. STEP #3: Combine the equations (one has to be negative) and be sure that you eliminated a variable. STEP #4: Solve for the remaining variable. STEP #5: Go back and plug in that value to solve for the other variable. STEP #6: Interpret your answer and write a sentence. Method Example Align the components of the system. 8x + 11y = 37 2x 11y = 7 Look for patterns (i.e. Least Common Multiple) to make one of the coefficients match the other, Combine the equations (one has to be negative) and be sure that you eliminated a variable, brow a line under the equations and add. Not necessary, because "11y" and 11y are opposites 8x + 11y = 37 2x 11y = 7 10x = 30 The result is an equation with only one variable, which we can solve. Thus, x =3 gives the xvalue of the solution to the given system. Go back and plug in that value to solve for the other variable. To find the yvalue, substitute 5 for x in either of the two equations of the system. X = 3 8x + 11y = 37 8(3) + 11y = y = 37 11y = y = 13 y = Interpret your answer and write a sentence. The solution to the system is 13 3, 11
7 Example 2: When you have to multiply: Method Align the components of the system. Look for patterns (i.e. Least Common Multiple) to make one of the coefficients match the other. Here, if we add the equations, no variables will be eliminated. However, if the y in the first equation were  2y, the yterms would be additive inverses and would be eliminated when added. We can use multiplication to multiply every member of the first equation by 2. Rewrite the system by replacing the first equation with our new equation from the step above. Combine the equations (one has to be negative) and be sure that you eliminated a variable. Go back and plug in that value to solve for the other variable. To find the yvalue, substitute 3 for x in either of the two equations of the system. Interpret your answer and write a sentence. Example 3x y = 8 x + 2y = 5 3x y = 8 2(3x y) = 2(8) 6x 2y = 16 6x 2y = 16 x + 2y = 5 6x 2y = 16 x + 2y = 5 7x = 21 x = 3 x + 2y = 5 (3) + 2y = 5 2y = 53 2y = 2 y = 1 The solution to the system is (3,1) Extra Hints for Substitution and Elimination Questions: When you solve using substitution and elimination, there are three possible outcomes. ANSWER GRAPH MEANING RESPONSE Inequality: i.e Parallel lines no points in common no solution A solution for both variables: i.e. (2, 4) Intersecting lines one point in common that point IS the solution Equality: i.e. 0 = 0 Same line all points in common infinitely many solutions
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