SAT Systems & Inequalities Cheat Sheet 2025 – Solve • Graph • Shade
Test Preparation
Jun 27, 2025
Master systems of equations and inequalities for the SAT Math section with essential tips, methods, and graphing strategies.
Want to ace the SAT Math section? Start here. Systems of equations and inequalities are key topics that make up about 30% of the SAT Math section. Here's what you need to know:
Systems of Equations: Solve for the point where two lines intersect using substitution, elimination, or graphing.
Inequalities: Define regions on a graph using boundary lines (dashed for strict inequalities, solid for inclusive ones) and shade the correct area.
Graphing Tips: Rewrite equations in slope-intercept form (y = mx + b) for clarity, plot points carefully, and verify solutions visually.
Key Methods: Use substitution when a variable is isolated, elimination for matching coefficients, and graphing for simpler equations or visual checks.
Mastering these skills boosts problem-solving speed and accuracy, helping you tackle tougher questions later. Let’s break it all down with clear examples and actionable strategies.
Since this article focuses on solving systems and inequalities the traditional way, you can also check out our comprehensive Desmos cheat sheet which shows you how to tackle these problems using Desmos.
Systems of Linear Inequalities in Two Variables - SAT Math
Substitution vs. Elimination: When to Use Each Method
When tackling a system of equations on the SAT, you have two primary tools at your disposal: substitution and elimination. Both methods aim to simplify the system into a single-variable equation, but knowing when to use each can save you precious time.
Understanding Substitution and Elimination
The substitution method involves isolating one variable in one equation and substituting its expression into the other equation. Let’s look at this example:
( y = 3x + 2 )
( 2x + y = 12 )
Here, the first equation already isolates ( y ). Substituting it into the second equation gives:
( 2x + (3x + 2) = 12 ),
which simplifies to ( 5x + 2 = 12 ).
On the other hand, elimination works by canceling out one variable through addition or subtraction of the equations. Sometimes, this requires multiplying one or both equations by a constant to align coefficients. Essentially, substitution replaces a variable with its equivalent expression, while elimination removes a variable altogether. Both methods ultimately produce a single-variable equation, allowing you to solve for the solution.
When to Use Each Method
Substitution is ideal when one variable is already isolated or can be isolated with minimal effort. For example:
If the equation looks like ( x = 4y + 1 ) or ( -3x + y = 7 ), substitution simplifies the process.
Elimination shines when the equations have identical or opposite coefficients. Consider these examples:
For ( 2x + 3y = 11 ) and ( 2x + 7y = 23 ), subtracting one equation from the other immediately eliminates ( 2x ).
Similarly, if the equations include opposite terms, like ( 2x + 2y = 7 ) and ( 5x - 2y = 14 ), elimination is a quick and efficient choice. It’s also effective when one equation contains a term that’s a multiple of the corresponding term in the other equation.
The key is to determine which approach requires less effort for the specific problem. Both methods are valid and will lead to the same solution.
David Ecsery, M.S. in Computer Science from California State University, East Bay, sums it up well:
"Convenience. That's all. If there's less work needed to solve one equation for one of the variables, substitute and simplify. If there's less work getting one equation lined up with the other - and, if needed, multiplying for addition or subtraction to eliminate a variable - do that."
Step-by-Step Decision Guide
When faced with a system of equations on the SAT, use these steps to decide the best method:
Examine the Equations:
Is one variable already isolated, or do the coefficients allow for easy cancellation? For example, if you see ( y = 2x - 5 ), substitution is the way to go. If the equations have matching coefficients, like ( 3x ) in both, or opposite terms, like ( +4y ) and ( -4y ), elimination will save time.
Play to Your Strengths:
Pick the method that feels more intuitive and reduces the amount of work required.
Both substitution and elimination are reliable strategies. The trick is to choose the one that simplifies the problem the most for you.
Graphing Tips for Systems and Inequalities
Graphing systems and inequalities is a powerful way to visualize solutions and intersection points. It’s especially helpful for double-checking answers from algebraic methods and can be a game-changer for those who understand concepts better through visuals.
How to Graph Systems of Equations
To graph any linear equation, start with the slope-intercept form: y = mx + b. Here, m is the slope (indicating how steep the line is), and b is the y-intercept (where the line crosses the y-axis). Converting equations into this form makes the graphing process much simpler.
Once in slope-intercept form, follow these steps:
Plot the y-intercept (b) on the graph.
Use the slope (m), expressed as "rise over run", to find additional points.
Connect the points with a straight line, extending it in both directions.
Repeat this process for each equation in the system. Be precise with your plotting, and ensure your graph has a clear, labeled scale. Even small mistakes in plotting can lead to incorrect solutions, so accuracy is key.
Finding Intersection Points
After graphing all the equations, focus on where the lines intersect:
One intersection point means there’s a unique solution.
No intersection (parallel lines) means there’s no solution.
Overlapping lines (same line) indicate infinitely many solutions.
To confirm, substitute the coordinates of the intersection point into both equations. This step ensures the solution satisfies both equations. Graphing makes it easy to spot these scenarios visually, often catching errors that algebraic methods might overlook.
Graphing Inequalities and Solution Regions
When graphing inequalities, begin by plotting the boundary line. Replace the inequality symbol with an equal sign temporarily to treat it like a regular equation.
The type of inequality determines how you draw the line:
Use a dashed line for strict inequalities (< or >), showing that points on the line are not included in the solution.
Use a solid line for non-strict inequalities (≤ or ≥), indicating that points on the line are included.
"A linear inequality occurs when the equal sign in a linear equation is replaced with an inequality symbol." - brainfuse.com
Next, determine which side of the boundary line to shade. Pick a test point that’s not on the line - (0,0) is often a convenient choice unless it lies on the line. Substitute the test point into the inequality:
If the inequality holds true, shade the region containing that point.
If it’s false, shade the opposite region.
Example 1: To graph y − 2x > 3, rewrite it as y > 2x + 3. Plot a dashed line for y = 2x + 3 and shade the region opposite the origin (0,0).
Example 2: For 3x + 4y ≤ 12, rewrite it as y ≤ (-3/4)x + 3. Plot a solid line for y = (-3/4)x + 3. Test the point (1, 1): 3(1) + 4(1) ≤ 12 simplifies to 7 ≤ 12, which is true. Shade the region containing (1, 1).
As a general guide:
For y < mx + b or y ≤ mx + b, shade below the line.
For y > mx + b or y ≥ mx + b, shade above the line.
Avoid common mistakes like using the wrong line style, testing boundary points, or shading the wrong region. Always double-check your test point calculations. Remember, the shaded area represents all the points that satisfy the inequality. Up next, explore the "Test a Point" method for even faster shading.
Test a Point Shading Method for Linear Inequalities
The test a point method is a straightforward way to figure out which side of a boundary line to shade when graphing linear inequalities. It removes the guesswork by verifying the solution region with just one test point.
What Is the Test a Point Method?
This method works hand-in-hand with graphing techniques to make solution regions crystal clear. After drawing the boundary line, the graph is divided into two regions. By choosing a single test point and plugging its coordinates into the inequality, you can determine which side of the line contains all the solutions.
How to Apply the Test a Point Method
Here’s how you can use this method step by step:
Pick a Test Point and Plug It In.
Choose a test point - (0,0) is usually a good choice unless it’s on the boundary line - and substitute its coordinates into the inequality.
Figure Out the Shading.
If the test point makes the inequality true, shade the half-plane containing that point.
If it’s false, shade the opposite half-plane.
Let’s break it down with an example. For the inequality 3x − 2y < 6:
Start by graphing the boundary line for 3x − 2y = 6. Use the intercepts (2,0) and (0,−3) to draw the line. Since the inequality is strict (<), use a dashed line to show the points on the line aren’t included in the solution.
Now, pick (0,0) as your test point and substitute it into the inequality:
3(0) − 2(0) < 6 simplifies to 0 < 6.
This is true, so shade the half-plane that includes (0,0).
Pro Tip: If you have time, double-check by testing another point from the shaded region to confirm your solution.
Common Mistakes to Avoid
Even a simple method like this can go awry if you’re not careful. Here are some common pitfalls:
Using a Test Point on the Boundary.
A point on the boundary won’t help you figure out which side to shade, as it satisfies the equation, not the inequality.
Wrong Line Style.
Use a dashed line for inequalities like < or >, and a solid line for ≤ or ≥. This ensures your graph reflects whether boundary points are included in the solution.
Forgetting to Flip the Inequality.
When solving for y, remember to reverse the inequality sign if you multiply or divide by a negative number. Missing this step can lead to shading the wrong region.
Misunderstanding the Test Point’s Role.
The test point is just a tool to identify the correct region. The solution set is the entire half-plane, not just the test point itself.
Arithmetic Errors.
Simple math mistakes during substitution can throw off your shading. Always double-check your calculations.
To be extra sure, test another point within the shaded region. This small step can save you from errors before finalizing your graph.
Quick Reference Summary Tables
Here’s a handy guide to key SAT strategies and common pitfalls. These tables pull together essential methods and rules to help you quickly review and reinforce what you've learned.
Substitution vs. Elimination vs. Graphing Comparison
Method | Best When | Quick Steps | |
|---|---|---|---|
Substitution | One equation is already solved for a variable (e.g., x = 4y + 1) or can be solved in one step | • Solve one equation for a variable | Great for equations like y = 3x + 2 |
Elimination | Both equations are in standard form (Ax + By = C) or have identical/opposite terms | • Align equations vertically | Best for matching coefficients |
Graphing | Coefficients are small, and solutions are integers | • Convert equations to slope-intercept form | Works well with simple coefficients |
As mathematician Jeff Suzuki explains:
"The quick answer is: if the equation is already solved for one variable, substitution is probably easier."
It’s important to remember that all systems of linear equations can be solved using either substitution or elimination. On test day, stick to the method that feels most natural to you.
Shading Rules Quick Reference
Inequality Symbol | Line Type | Test Point Method | What to Remember |
|---|---|---|---|
< or > | Dashed line | Pick (0,0) unless it’s on the boundary | Boundary points are excluded |
≤ or ≥ | Solid line | Use the same test point process | Boundary points are included |
Multiple Inequalities | Overlap shaded regions | Test each inequality separately | Find the intersection of regions |
Shading Checklist:
Confirm if your test point lies on the boundary line (if it does, choose a different point).
Check your line style - use a dashed line for strict inequalities and a solid line for inclusive ones.
When solving for y, don’t forget to flip the inequality sign if you multiply or divide by a negative number.
The test point method eliminates guesswork. Instead of trying to visualize which side to shade, let the math guide you to the correct solution.
These tables are your go-to tools for choosing the best approach during the SAT. Keep them in mind as you tackle problems on test day!
Conclusion and Next Steps
Key Strategies Review
You’ve now got a solid foundation for tackling systems and inequalities in the SAT's Heart of Algebra section. Since this section makes up 33% of the SAT math portion, mastering these skills can have a big impact on your overall score.
Here’s what to focus on:
Linear Equations and Systems: Get comfortable solving and interpreting these.
Graphing: Use the test point method to simplify graphing problems. Start with (0,0) unless the line passes through the origin. Remember: dashed lines mean boundary points are excluded, while solid lines include them.
Linear Inequalities: Memorize key rules and formulas, including the slope-intercept form (y = mx + b), as these are essential for solving more advanced problems.
These strategies are your foundation. Now, it’s time to put them into practice.
How to Practice These Skills
To sharpen your skills, focus on targeted and consistent practice. This will help you build both accuracy and speed over time.
Work step-by-step: Slow and careful problem-solving reduces errors.
Plug in numbers: Test answer choices with specific values, but don’t rely only on simple ones like –1, 0, or 1.
Graphing tips: Always label your axes and clearly mark shading on your scratch paper. Use the test point method to double-check your work.
For extra support, take advantage of tools like Khan Academy and Desmos. These resources offer guided lessons and interactive graphing practice to help you reinforce what you’ve learned.
Finally, don’t rush. Take your time with multi-step problems. Focus on getting the answer right first - speed will naturally follow as you grow more confident with these techniques.
FAQs
When should I use substitution or elimination to solve a system of equations on the SAT?
When deciding between substitution and elimination, it all comes down to how the equations are structured. Go with substitution if one equation already has a variable isolated or can be easily rearranged to isolate one. This method is especially handy when the coefficients are straightforward or when one variable is neatly expressed in terms of the other.
On the other hand, elimination is the better choice when the coefficients of one variable are already opposites - or can be made opposites with just a little adjustment. This method shines when you want to align the terms and quickly cancel out one variable, making it easier to solve for the other. The key is to pick the approach that simplifies the problem the most, saving you valuable time during the test.
What are some common graphing mistakes with inequalities, and how can I shade correctly?
When graphing inequalities, there are a few common pitfalls to watch out for. These include misdrawing the boundary line, misinterpreting inequality symbols (like shading the wrong region), and skipping the crucial test point step. To graph inequalities correctly, follow these key guidelines:
Draw the boundary line properly: Use a solid line for inequalities like ≤ or ≥, and a dashed line for < or >. This distinction shows whether points on the line are included or excluded.
Use a test point: Pick a simple point, such as (0, 0) (as long as it’s not on the line), to determine which side of the line satisfies the inequality. If the test point works in the inequality, shade that side; if not, shade the other side.
Handle negative numbers carefully: If you multiply or divide by a negative number while solving, don’t forget to flip the inequality sign - this will change which side of the line gets shaded.
By sticking to these steps, you’ll ensure your graphing and shading are spot on every time!
How does the 'Test a Point' method work for shading regions in linear inequalities?
The 'Test a Point' Method
The 'Test a Point' method offers a simple way to determine which side of the boundary line to shade when graphing linear inequalities. Here's how it works:
Choose a point that isn't on the boundary line - most people go with (0, 0) because it's easy to calculate, as long as the line doesn't pass through it. Plug the coordinates of that point into the inequality.
If the inequality holds true, shade the side of the line where the point is located.
If it doesn’t, shade the opposite side.
This technique helps you identify the correct region quickly, making the solution to the inequality much clearer.