SAT Math questions frequently test students' ability to work with systems of equations, generally by providing two different equations with two different variables and asking students to solve for one or more of the variables. While these questions may at first appear daunting, the good news is that you can easily solve the majority of them by simply graphing the equations in the SAT's built-in Desmos app.
Using Desmos to solve systems of equations questions
To see how this works, check out the typical systems of equations question below.
Entering the two equations in Desmos results in this graph:
In the graph, the two lines intersect at the point (10, 10). so the x value of the solution is 10. That's your answer—easy peasy!
Note that this approach will also work for similar questions that test non-linear equations. Additionally you don't need to rearrange your equations to set them equal to y before you enter them in Desmos; just enter them as they're given. No calculations required!
Solving systems of equations that have infinite or no solutions
More challenging SAT systems of equations questions will provide you with a system of equations that has infinite solutions or no solutions, and one of the equations will have a constant in place of a number. In order to answer these questions, you will need to solve for the constant first, even if the goal of the question is to find the value of x or y. Before you can do this though, you will need to know what it means for a system of equations to have infinite or no solutions.
Infinite solutions
If a system of equations has infinite solutions, then for any given value of x, the two equations will have the same y value. This means that if you graph the two equations, their lines will overlap completely, intersecting an infinite number of times. Take for example these two equations:
Entering them in Desmos results in this graph:
The graph includes only a single line because the graphs of the two equations overlap completely.
The reason that they overlap completely is that what has been given is really one equation, not two separate equations. Note what happens when you multiply the first equation by two:
Now distribute:
As you can see, the two equations are now identical.
Understanding that if a system of equations has infinite solutions then the given equations are identical (after one of the equations is multiplied or divided by a number) is crucial for solving many SAT questions. Here's an example of how you might see this concept test on the SAT:
Here's how to solve it.
Step 1: Move all of the terms with variables to one side of the equation, and move all of the numbers to the other side of the equation.
In this case, we will need to add 5x to each side of the second equation.
Simplify:
Step 2: Find the multiplier.
Now that the two equations are in the same form, it's much easier to compare them. Because the question states that the system of equations has no solution, you know that the numbers in the terms on the left side of the equations will be in a constant ratio.
To find the multiplier, you will need to compare the 36x term in the top equation with the 12x term in the bottom equation. You cannot compare the terms that include y, because you do not know the value of c, and you cannot use the terms on the right side of the equation because those numbers will not necessarily have any relationship to each other.
The result of dividing 36x by 12x is 4, so you know that your multiplier is 4.
Step 3: Multiply
Multiply the second equation by 4:
Simplify:
Step 4; Solve
Looking at the two equations, you now know that —78y = 3cy, or —78 = 3c. Divide both sides by 3 to find that c = —26. That's it!
SAT systems of equations questions certainly require care and precision to be solved correctly. However, with ample practice and an eye for detail, you can undoubtedly master this question type on test day!
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