Systems
What is a system??? A system is two or more equations that use the same variables (usually x and y) that you solve at the same time.
The system's solution (answer) looks like an ordered pair (x,y). This represents where the lines cross or intersect on a graph.
The system's solution (answer) looks like an ordered pair (x,y). This represents where the lines cross or intersect on a graph.
Getting an equation into y=mx+b form
1. Add or subtract the x to the other side
2. Divide the number attached to the y (do not put any decimals here, simplify a fraction as much as possible
3. If needed, switch the right side around to make y=mx+b
2. Divide the number attached to the y (do not put any decimals here, simplify a fraction as much as possible
3. If needed, switch the right side around to make y=mx+b
Types of Solutions
ONE SOLUTION: This happens with the slope and y intercepts are different and the lines intersect on the graph.
INFINITELY MANY SOLUTIONS: This happens when the slope and y intercepts are the exact same and the 2 lines on the graph appear as one.
NO SOLUTION: This happens when the slope is the same but the y intercepts are different, resulting in parallel lines on a graph.
INFINITELY MANY SOLUTIONS: This happens when the slope and y intercepts are the exact same and the 2 lines on the graph appear as one.
NO SOLUTION: This happens when the slope is the same but the y intercepts are different, resulting in parallel lines on a graph.
solving with graphing
1. First, get the two equations into y=mx+b form (if needed, see above).
2. Find the y intercept (the number in the 'b' spot of y=mx+b) and graph this point on the y axis (vertical line)
3. Find the slope (the number in the 'm' spot of y=mx+b). This will tell you rise over run. If it is a fraction, go up or down from the y intercept point and over to the right the number of spaces it tells you to move. If it is a whole number such as 3, make it into a fraction by putting the number over a one, such as 3/1. Now you have a fraction to rise and run from the y intercept.
4. Repeat the previous steps for the other equation(s).
5. Look at your graph and see where the two lines intersect. This is your solution so write it as an ordered pair (x,y). If the lines do not cross, see your other options for solutions above.
2. Find the y intercept (the number in the 'b' spot of y=mx+b) and graph this point on the y axis (vertical line)
3. Find the slope (the number in the 'm' spot of y=mx+b). This will tell you rise over run. If it is a fraction, go up or down from the y intercept point and over to the right the number of spaces it tells you to move. If it is a whole number such as 3, make it into a fraction by putting the number over a one, such as 3/1. Now you have a fraction to rise and run from the y intercept.
4. Repeat the previous steps for the other equation(s).
5. Look at your graph and see where the two lines intersect. This is your solution so write it as an ordered pair (x,y). If the lines do not cross, see your other options for solutions above.
Solving with substitution
1. Make sure at least one of the equations start with a y= or an x=
IF THE EQUATION STARTS WITH A Y=
Use the other equation that does not start with a y= and find where y is int eh equation.
Replace the y with the contents of the y= equation and solve for x. There will not be any y's in this step once you replace them. ONLY X's SHOULD APPEAR.
You will now have x= a number Take the number you just found and plug it into one of the equations (it does not matter which one you plug the number into). Solve for y.
Write the solution as an ordered pair (x,y)
IF THE EQUATION STARTS WITH A X=
Use the other equation that does not start with a x= and find where y is int eh equation.
Replace the x with the contents of the x= equation and solve for y. There will not be any x's in this step once you replace them. ONLY Y's SHOULD APPEAR.
You will now have y= a number Take the number you just found and plug it into one of the equations (it does not matter which one you plug the number into). Solve for x.
Write the solution as an ordered pair (x,y)
IF THE EQUATION STARTS WITH A Y=
Use the other equation that does not start with a y= and find where y is int eh equation.
Replace the y with the contents of the y= equation and solve for x. There will not be any y's in this step once you replace them. ONLY X's SHOULD APPEAR.
You will now have x= a number Take the number you just found and plug it into one of the equations (it does not matter which one you plug the number into). Solve for y.
Write the solution as an ordered pair (x,y)
IF THE EQUATION STARTS WITH A X=
Use the other equation that does not start with a x= and find where y is int eh equation.
Replace the x with the contents of the x= equation and solve for y. There will not be any x's in this step once you replace them. ONLY Y's SHOULD APPEAR.
You will now have y= a number Take the number you just found and plug it into one of the equations (it does not matter which one you plug the number into). Solve for x.
Write the solution as an ordered pair (x,y)
Solving with addition
1. Make sure you have either x's or y's that are the exact opposite (one is negative and the other is positive).
2. Add all pieces of the two equations together. The x's or y's that were opposites will cancel out and you will be left with one variable = a number.
3. Solve for the remaining variable (normally you just divide)
4. Now you have x = a number OR y= a number. Substitute the number you just found back into one of the original equations (it does not matter which one you choose).
5. Solve for the other variable.
6. Write the solution as an ordered pair (x,y)
2. Add all pieces of the two equations together. The x's or y's that were opposites will cancel out and you will be left with one variable = a number.
3. Solve for the remaining variable (normally you just divide)
4. Now you have x = a number OR y= a number. Substitute the number you just found back into one of the original equations (it does not matter which one you choose).
5. Solve for the other variable.
6. Write the solution as an ordered pair (x,y)
Solving with subtraction
1. Make sure that you have either x's or y's that are the exact same (both are negative OR both are positive).
2. Subtract all pieces of the two equations together. The x's or y's that were the same will cancel and you will be left with the other variable = a number
3. Solve for the variable left in the previous step.
4. Now you have x= a number OR y= a number. Substitute the number you just found back into one of the original equations (it does not matter which one)
5. Solve for this variable
6. Write the solution as an ordered pair (x,y)
2. Subtract all pieces of the two equations together. The x's or y's that were the same will cancel and you will be left with the other variable = a number
3. Solve for the variable left in the previous step.
4. Now you have x= a number OR y= a number. Substitute the number you just found back into one of the original equations (it does not matter which one)
5. Solve for this variable
6. Write the solution as an ordered pair (x,y)