Overview
A system of equations refers to at least 2 equations that have at least one variable in common. Each equation gives a piece of information about the variables that it contains. One way the variables can be solved is by using substitution.
Solving Systems of Equations with One Variable
Suppose that a system of two equations has one variable in common, such as x + 2 = 3 and 2x + 10 = 12. To see if the systems have a solution in common, solve one of the equations and use substitution to solve the other. The equation x + 2 = 3 can be solved by subtracting the additive inverse of 2 from both sides of the equation, such that x + 2 – 2 = 3 – 2 or x = 3 – 2 or x = 1. Then 2∙1 + 10 = 12.
Figure 1: Subtracting the additive inverse of 11 from both sides of the equation.
Solution of a System of Equations in Two Variables
Suppose that two equations in a system have two variables in common. For example, x +y =8 and 2x – y = 1. Similar to systems of equations with one variable, one equation can be solved and then the values of each variable can be substituted in the other equation. If x equals 1, then y equals 7; if x equals 2, y equals 6; if x equals 3, then y equals 5; and if x equals 4, then y equals 4. Suppose that the first pair 1 and 7 is tried. The solution fits the first equation, but 2 – 7 is equal to -5, not 1. If x equals 7 and y equals 1, then 14 -1 =13, which is not equal to 1. If x = 2 and y = 6, then 4 – 6 = -2, and if x = 6 and y = 2, then 12 – 1 is 11, which does not equal 1. If x = 3 and y = 5, then 6 – 5 is 1, and the system is solved using substitution.
Figure 2: Solving systems of equations by substitution.
More Substitution to Solve Equations
In the previous method, one equation was solved and then the values for the variables were substituted in the other equation. Another method calls for setting one equation as equal to one of the variables and then substituting that value in the second equation. This method eliminates some of the trial and error, if the systems of equations have a single data point that intersects. Suppose that x + y = 8 and 2x – y = 1 as before. Then y = 8 – x and 2x – 8 + x = 1. The second equation then becomes 3x = 1 + 8, or 3x = 9, x = 3. Then y = 8-3 so y = 5. The ordered pair to solve the system is (3, 5).
Graphing the System of Equations in Two Variables
If the equations in a system of equations are in two variables, they can be graphed as sets of ordered pairs. The x axis is conventionally horizontal and the y axis is conventionally vertical. Each data point on the linear equation x + y = 8 is represented by the ordered pairs on one line, and each data point on the linear equation 2x – y = 1 is represented by ordered pairs on another line. The common solutions are represented by the single point (3, 5).
Figure 3: Graphing systems of equations that have a single solution in common.
Interested in algebra tutoring services? Learn more about how we are assisting thousands of students each academic year.
SchoolTutoring Academy is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Jamestown, RI: visit Tutoring in Jamestown, RI