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When solving linear systems, the elimination method - sometimes called the addition method - usually is the method of choice. This technique is completely algebraic and quick once you get the hang of it. The idea is to eliminate one variable by adding equivalent equations together.

Solving Linear Systems Playlist on YouTube

Solve the system using the elimination method:

Step 1: Multiply one or both of the equations by factors that will line up one variable to eliminate.

Step 2: Add the equations together.

Step 3: Back substitute and present the answer as an ordered pair.

Tip: If you multiply an equation by any number - remember to distribute!

Solve the system using the elimination method:

To eliminate the variable y, multiply the first equation by 3 and the second equation by 2.

Now add the equations together.

You will always have to back substitute to find the value of the other coordinate.

Solve the systems using the elimination method:

You will likely encounter systems that are not lined up in standard form. In this case, you should first rearrange the equation before using the elimination method.

Clearing Fractions: If we are given an equation with fractional coefficients, we can clear them out by multiplying both sides by a common multiple of the denominator. This is a handy technique which we will use often in our study of Algebra. Do not abuse this, as it only works on equations and not expressions.

The LCM of the denominators is 30.

Distribute and then simplify.

No more fractions; now that is nice!

This gives equivalent equations.

Multiply both sides of any equation by the LCD to clear the fractional coefficients.

Solve the systems using the elimination method.

Video Examples on YouTube:

The substitution method for solving linear systems is a completely algebraic technique. There is no need to graph the lines unless you are asked to. This method is fairly straight forward and always works, the steps are listed below.

Solving Linear Systems Playlist on YouTube

Solve the system using the substitution method:

Step 1: Use either equation and solve for a variable.

In this case, we solved the first equation for y.

Step 2: Substitute the resulting quantity into the other equation.

Here we substituted the quantity found for y into the second equation.

Step 3: Solve for the remaining variable.

Step 4: Back substitute to find the value for the other variable.

Step 5: Present your answer as an ordered pair (x, y).

It does not matter if you choose to solve for x or y first. However, make sure that you do not substitute into the same equation in step 2.

Remember that we are trying to find the simultaneous solutions or the points where the two lines intersect. Next we will see what happens when the system is dependent, in other words, when the system consists of two lines that are the same.

Solve the system using the substitution method:

Solve for y in the first equation.

Any true statement, including 0 = 0, indicates a dependent system.

The next system consists of two parallel lines which has no simultaneous solution.

Solve the system using the substitution method:

Solve for x in the first equation.

Any false statement indicates an inconsistent system.

Solve the systems using the substitution method.

Typical word problem: When Joe walked away from the craps table he had 45 chips. He had a combination of $5 and $25 chips that added to a total of $625. How many of each chip did he have?