SAT Math Preparation: Tackling Systems of Equations

The SAT Math section assesses a student’s ability to problem-solve, and among the critical topics tested, systems of equations stand out as both fundamental and often daunting. However, with the right techniques and understanding, they can be simplified. This article aims to break down the methods to solve systems of equations efficiently for the SAT.

1. What is a System of Equations?

A system of equations is a set of two or more equations with two or more variables. The goal is to find the values for the variables that make all the equations true simultaneously. Most commonly on the SAT, you’ll be dealing with systems of linear equations in two variables.

2. Methods to Solve Systems of Equations:

a) Substitution:

This method involves solving one equation for one variable in terms of the other variables. Once this is done, this expression is substituted into the other equation(s).

Steps:

1. Solve one equation for one variable.
2. Substitute this expression into the other equation.
3. Solve the resulting equation.
4. Plug the solution from step 3 into one of the original equations to solve for the other variable.

Example:

Given the system:

1. y=2x+3
2. 3x+y=12

From (1), you already have y in terms of x. Substitute this into (2):

3x+2x+3=12

This simplifies to 5x=9 or x=1.8. Using this value in (1), we get y=6.6.

b) Elimination (or Addition/Subtraction method):

This method involves adding or subtracting the equations to eliminate one variable, allowing you to solve for the other variable.

Steps:

1. Multiply one or both equations by a number(s) such that the coefficients of one variable are opposites.
2. Add or subtract the equations to eliminate one variable.
3. Solve for the remaining variable.
4. Substitute the solution from step 3 into one of the original equations to find the other variable.

Example:

Given the system:

1. 2x−y=8
2. x+y=3

By adding the two equations, y is eliminated:

3x=11 or x=11/3​. Using this value in (2), we get y=−2/3​.

3. Tips for the SAT:

• Decide on a method: If one equation is already solved for one variable, consider substitution. If the coefficients of one variable are opposites or can easily be made opposites, consider elimination.
• Avoid messy arithmetic: The SAT often provides numbers that work out neatly. If your calculations are getting overly complicated, double-check your steps.
• Draw on the answer choices: Sometimes, plugging in the provided answer choices can be a quick way to solve the system.

4. Systems of Equations in Word Problems:

The SAT frequently presents systems of equations in word problem format. Here’s a strategy:

1. Assign variables to unknown quantities.
2. Translate words into mathematical expressions. Phrases like “total of,” “difference between,” and “product of” can help set up the system.
3. Set up and solve the system using either substitution or elimination.

Conclusion:

Systems of equations may seem challenging at first, but with consistent practice and strategic approaches, you can master them for the SAT Math section. Remember, understanding the logic behind each step is vital, so ensure you’re not merely memorizing procedures but truly grasping the concepts. As with all SAT preparation, practice is key. Happy studying!