SAT Math: Heart of Algebra — Linear Equations, Linear Functions and Systems of Equations

TL;DR

You'll need to solve linear equations, understand how lines behave on a graph, and find where two lines cross. It's all about finding unknown values that make statements true. These skills are fundamental for about a third of the SAT Math section.

1. The Mental Model

Think of linear equations as simple balancing acts where you're trying to find a missing piece. Linear functions describe steady, straight-line changes. Systems of equations are like solving two balancing acts at once to find the one pair of numbers that works for both.

2. The Core Material

What's a Linear Equation?

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A linear equation is a statement like 2x + 3 = 7 where x is some unknown number. Your goal is to isolate the variable (get x by itself) to find its value. Remember, whatever you do to one side of the equals sign, you must do to the other to keep it balanced.

Here's how you solve for x:
1. 2x + 3 = 7
2. Subtract 3 from both sides: 2x = 4
3. Divide both sides by 2: x = 2

What's a Linear Function?

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A linear function describes a straight line on a graph. The most common form is slope-intercept form: y = mx + b.
* y and x are your variables (points on the line).
* m is the slope. It tells you how steep the line is and its direction (rise over run). A positive m means the line goes up to the right; a negative m means it goes down to the right.
* b is the y-intercept. This is where the line crosses the y-axis (when x = 0).

You might also see standard form: Ax + By = C. You can usually convert this to slope-intercept form by isolating y.

Systems of Linear Equations

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This is when you have two or more linear equations and you need to find the (x, y) pair that works for all of them. Graphically, this is where the lines intersect.

There are three main methods to solve systems:

1. Substitution:

   • Solve one equation for one variable (e.g., y = 2x + 1).
   • Substitute that expression into the other equation.
   • Solve for the remaining variable.
   • Plug that value back into one of the original equations to find the first variable.

2. Elimination (or Linear Combination):

   • Multiply one or both equations by a number so that when you add or subtract the equations, one variable cancels out.
   • Solve for the remaining variable.
   • Plug that value back into one of the original equations to find the first variable.

3. Graphing (less common for SAT, but good for understanding):

   • Graph both lines.
   • The point where they cross is your solution (x, y).

What kind of solutions can you get?
* One solution: The lines intersect at a single point (most common).
* No solution: The lines are parallel and never intersect (they have the same slope but different y-intercepts). Example: y = 2x + 3 and y = 2x - 1.
* Infinitely many solutions: The lines are actually the same line (they have the same slope and the same y-intercept). Example: y = 2x + 3 and 2y = 4x + 6.

Here's a quick look at the types of solutions for systems of equations:

graph TD
    A["System of Two Linear Equations"] --> B{"Do the lines intersect?"}
    B -- "Yes, at one point" --> C["One Solution (x, y)"]
    B -- "No, they're parallel" --> D{"Are they the SAME line?"}
    D -- "Yes" --> E["Infinitely Many Solutions"]
    D -- "No, different y-intercepts" --> F["No Solution"]

3. Worked Example

Let's solve the following system of equations:
1. 3x + 2y = 12
2. y = x - 1

I'll use the Substitution Method since the second equation already has y isolated.

Step 1: Substitute
Substitute (x - 1) for y in the first equation:
3x + 2(x - 1) = 12

Step 2: Solve for x
3x + 2x - 2 = 12
5x - 2 = 12
5x = 14
x = 14/5 or x = 2.8

Step 3: Solve for y
Now that we have x, plug it back into either original equation. The second one is easier:
y = x - 1
y = 14/5 - 1
y = 14/5 - 5/5
y = 9/5 or y = 1.8

Solution: The solution is (14/5, 9/5) or (2.8, 1.8).

4. Key Takeaways

• When solving linear equations, remember to keep both sides balanced by doing the same operation to each.
• Slope (m) tells you the rate of change and direction of a line, while the y-intercept (b) is the starting value.
• The solution to a system of equations is the point (x, y) where the lines intersect.
• Be ready to use either substitution or elimination to solve systems of equations quickly.
• Understand that systems can have one solution, no solution (parallel lines), or infinitely many solutions (the same line).

Common Mistakes to Avoid:
* Forgetting to distribute a negative sign when using elimination or substitution.
* Mixing up x and y coordinates, especially for intercepts.
* Making calculation errors when adding, subtracting, multiplying, or dividing integers or fractions.
* Assuming lines are parallel just because they look like it on a quick sketch; always check the slopes.

5. Now Try It

Take this system:
Equation 1: 2x - y = 7
Equation 2: 4x + 3y = 7

Solve for x and y. What should success look like? You should end up with specific, single values for x and y that make both equations true when you plug them back in. Aim to finish this within 5 minutes.