Solving Systems of Equations Guide PDF

How to Solve Systems of Equations A system of equations is two or more equations that share/have the same variables (an unknown number such as x or y). The goal is to nd the values of x or y that make both equations true. Systems of equations are often written as: These systems can be solved usi...

How to Solve Systems of Equations A system of equations is two or more equations that share/have the same variables (an unknown number such as x or y). The goal is to nd the values of x or y that make both equations true. Systems of equations are often written as: These systems can be solved using each other. Substitution vs. Elimination There are two main ways to solve systems of equations: Substitution: Replace one variable with an expression from another equation. Elimination: Add or subtract the equations to remove (“eliminate”) one variable. Example Problem: Substitution We have two equations in a system: {2x + y = 10 x+y =6 Because we can isolate either the x or y variable in the first equation, we will solve the system using the substitution method. fi  Step 1: “Isolate” a Variable Start with one equation and solve for one variable (like x). Example: If x + y = 6, solve for x: x+y= -y -y Subtract y from both sides to isolate x by itself x=6-y Step 2: Substitute x Variable to Solve for y Variable We know that x = 6 - y, now solve for y using a second equation 2x + y = 10 2 (6 - y) + y = 10 First, plug in x = 6 - 1 for x variable. *Note: What do parentheses ( ) mean When a number is outside parentheses, we multiply everything inside the parentheses by the number outside. 2 (6 - y) = 2 x 6 and 2 x y Multiply everything inside the parenthesis by the number outside 12 - 2 y 6. ?. : Next, rewrite the equation as: 12 - 2y + y = 10 Next, isolate the variable y on one side of the equation. 12 - 2y + y = 10 - 12 -12 Subtract -12 from both sides of the equation to isolate the y variables. Step 3: Solve for the Remaining Variable The equation will now be written as: -2y + y = - 2 -2y and y can be combined because they have the same variables. 2y + y = - y So, - y=-2 To get rid of the negative sign in front of the y we divide - 1 on both sides. - y=-2 / -1 /-1 y=2 **Now that we know y = 2, we can substitute this into the other equation to find x.**  Step 4: Final Step: Solve for x Now that we know y = 2, substitute it into the first equation to find x: x+y= Substitute y = 2: x+2= - 2 -2 Subtract 2 from both sides to isolate x. x= 4 Solution: The final solution to the system is x = 4 and y = 2 Remember these steps: 1. Solve for one variable. 2. Substitute into the second equation. 3. Solve for the remaining variable. 4. Plug back in to nd the rst variable. 5. Check your answers Tips for Success Focus on one variable at a time. Keep equations balanced by doing the same thing to both sides. Write neatly and double-check your steps. 6 6 fi. fi