Solving systems of equations is a important problem solving skill that is being asked for more and more in many different fields.
Math teachers often use system solving as part of their curriculum, and students are required to solve them at least occasionally. Even outside of the education field, employers are looking for people with this skill.
Solving a system of equations means finding out what values or variables go together. For example, if there is a equation Y = 1 + X, then the solution is that X = 1.
Once you figure out how to solve one type of system of equations, you can apply this to all other systems! Luckily, there is an easy way to do this that will be explained in this article.
Find the x-coordinate of the intersection
Once you have found the y-coordinate of the intersection, your next step is to find the x-coordinate of the intersection. This is done by solving either equation for x and then plugging this value into the other equation.
For example, if equation one yields a x-coordinate of 2 and equation two yields a x-coordinate of –3, then the intersection lies at 2, –3.
It is important to note that if one of the equations yields a negative x-coordinate, then the point of intersection is on the opposite side of the graph. This is because both equations are being solved for x, so only one will be correct.
Solving for only one variable in an equation will usually result in an incorrect answer.
Set y = x and solve for x
To solve for x when y = x, you must find the value of x that makes the two equations equal. This is done by substituting x for y in one equation and solving for x.
Then, plug this value of x into the other equation to find y. If you do this correctly, both equations will have the same value of x and y.
Here is an example of solving for the solution to the system of equations 1 − y = 2x + 3 and y = 2/3. First, substitute 2/3 for y in the first equation to get 1 − (2/3) = 2x + 3, which equals −1.
Then, solve for the unknown variable, x, by subtracting 1 from both sides. The final step is to plug -1 into the second equation to find that the solution to the system of equations is x = 2.5.
Set y = x + 4 and solve for x
In this case, you would set y equal to x + 4, or what x would be when x + 4 is equal to 1.5x – 4. This would be x + 4 = 1.5(x – 4), solve for x, and then take the original x and add +4.
This would give you the solution of what y equals when x equals 2. Therefore, the solution to this system of equations is that y equals 1.5x + 1 and x equals 2.
You could have also solved this by setting y equal to –x and solving for x. The solution would be the same, however!
Beware that some computers do not compute the same answer for these solutions due to rounding errors.
Check to make sure the solution makes sense
Once you have your solution, it is important to check to see if it makes sense. Does each value of X correspond to a value of Y?
To do this, put one of the solutions for X in the Y=1 equation and calculate the corresponding Y. Then, put one of the solutions for X in the Y=–X equation and calculate the corresponding Y. Do these values of Y make sense?
For example, if you solved for X as 2, then calculated Y as 1.5*2–4=−2, which does not make sense. This means that 2 is not a valid solution for X! Make sure to check all possible solutions to ensure that none are invalid.
Solving systems of equations is an easy way to learn how to solve for unknowns using logic. Practicing more problems will improve your speed and accuracy.
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