SAT Math: Heart of Algebra 📋

SAT Math Overview

The Heart of Algebra is the single largest topic on the SAT Math section, accounting for 19 out of 58 questions (about 33%). Eight of those appear in the no-calculator portion and 11 in the calculator portion. They can show up as either multiple-choice or grid-in questions.

This topic tests your ability to work with linear equations, inequalities, and functions. You'll need to solve them fluently, build systems of equations, and connect equations to their graphs on the coordinate plane. The guide below covers all four skill areas College Board groups under Heart of Algebra.

📗 Main Heart of Algebra Topic Areas

College Board organizes Heart of Algebra into four skill groups:

📏 Linear Equations, Linear Inequalities, and Linear Functions in Context

• Translate real-world situations into algebraic expressions, equations, or inequalities. Define your variables, set up the math, solve, and then interpret your answer back in context.

🔄 Systems of Linear Equations and Inequalities in Context

• Create and solve systems of equations or inequalities that model a situation with more than one unknown.

🏋️‍♂️ Fluency in Solving Linear Equations, Linear Inequalities, and Systems of Linear Equations

• Solve equations, inequalities, and systems efficiently and accurately. These questions may be purely algebraic with no real-world context.

📋 The Relationships among Linear Equations, Lines in the Coordinate Plane, and the Contexts They Describe

• Connect equations to their graphs. Determine whether a system has one solution, no solution, or infinitely many solutions by analyzing slopes and intercepts or by graphing.

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📏 Linear Equations, Linear Inequalities, and Linear Functions in Context

This skill group is about turning a real-world scenario into algebra, solving it, and interpreting the result.

🧠 What You Need to Know: Linear Equations, Inequalities, and Functions in Context

Know the difference between expressions, equations, and inequalities:

• Linear expressions contain variables and constants but have no equal sign or inequality symbol. Examples: $5x - 3$, $2y - 7$.
• Linear equations set two expressions equal to each other using an equal sign. Example: $5x - 3 = 12$, which has the solution $x = 3$.
• Linear inequalities use inequality symbols ($<, >, \leq, \geq$) instead of an equal sign. Example: $2x + 1 > 5$, which gives the solution range $x > 2$.

The key distinction: expressions don't have an equal sign or inequality symbol. Equations give you a single solution. Inequalities give you a range of solutions.

What you need to be able to do:

• Define variables that represent the quantities described in the problem.
• Build an expression, equation, or inequality from the relationships in the problem.
• Solve the equation or inequality, which may require multiple steps (distributing, combining like terms, etc.).
• Interpret your answer in context. The SAT often asks what does your answer mean in this situation?

🤓 Applying Your Knowledge: Linear Equations, Inequalities, and Functions in Context

The following question is from the 2020 SAT Study Guide (all credit to College Board).

Interpreting Linear Inequalities Practice

To edit a manuscript, Miguel charges $50 for the first 2 hours and $20 per hour after the first 2 hours. Which of the following expresses the amount, C, in dollars, Miguel charges if it takes him x hours to edit a manuscript, where x > 2?

A) C = 20x

B) C = 20x + 10

C) C = 20x + 50

D) C = 20x + 90

The correct answer is B.

Here's how to work through it:

1. Miguel charges a flat $50 for the first 2 hours.

2. After those 2 hours, he charges $20 per hour. The number of additional hours beyond 2 is $(x - 2)$.

3. Combine both parts: $C = 50 + 20(x - 2)$

4. Distribute: $C = 50 + 20x - 40$

5. Simplify: $C = 20x + 10$

A common mistake here is writing $20x + 50$, which would mean Miguel charges $20 for all x hours on top of the $50 flat fee. But the $50 already covers the first 2 hours, so you need to subtract those 2 hours from the hourly portion.

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🔄 Systems of Linear Equations and Inequalities in Context

This skill group focuses on problems where you have more than one unknown and need to set up multiple equations or inequalities to model the situation.

🧠 What You Need to Know: Systems of Equations and Inequalities in Context

Know the difference between systems of equations and systems of inequalities:

• A system of equations is two or more equations with the same variables that must all be true at the same time. The solution is the specific set of values that satisfies every equation. Example: $2x + 3y = 10$ and $3x - 2y = 4$.
• A system of inequalities is two or more inequalities with the same variables. Instead of one specific solution, you get a region of values that satisfies all the inequalities simultaneously. Example: $2x + y \leq 8$ and $x - 3y > 2$.

How to decide which one to set up:

• Use a system of equations when the problem describes exact relationships and you need a specific answer (e.g., "how many of each item did she buy?").
• Use a system of inequalities when the problem describes constraints or limits and asks for a range of possibilities (e.g., "what combinations are possible given these budget limits?").

What you need to be able to do:

• Define multiple variables based on the problem's context.
• Translate the relationships and constraints into a system.
• Solve the system and interpret the solution in context.

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🏋️‍♂️ Fluency in Solving Linear Equations, Linear Inequalities, and Systems of Linear Equations

This skill group is about raw solving ability. Questions here may have no real-world context at all. They can appear in either the calculator or no-calculator section, so practice solving by hand.

🧠 What You Need to Know: Solving Equations, Inequalities, and Systems

🤔 Tips for Solving Linear Equations

• Isolate the variable. Your goal is to get the variable alone on one side.
• Do the same thing to both sides. Whatever operation you perform on one side, do it to the other to keep the equation balanced.
• Combine like terms first. Simplify each side before you start moving things around.
• Clear fractions or decimals early. Multiply both sides by the least common denominator to work with whole numbers instead.
• Plug in answer choices. If you're stuck on a multiple-choice question, substitute each answer into the equation and see which one works.
• Check your answer. Substitute your solution back into the original equation to verify it. This is especially important on grid-in questions where you can't check against answer choices.

💡 Tips for Solving Linear Inequalities

All the equation-solving tips above still apply, with these additions:

• Flip the inequality sign when multiplying or dividing by a negative number. This is the most common mistake on inequality problems. For example, if you divide both sides of $-3x > 12$ by $-3$, the result is $x < -4$ (the sign flips).
• Know your symbols. Make sure you're clear on the difference between strict inequalities ($<, >$) and inclusive ones ($\leq, \geq$).
• Test a value from your solution range. Pick a number that falls in your answer range and plug it back into the original inequality to confirm it works.

✨ Tips for Solving Systems of Equations

You have two main methods. Practice both so you can pick the faster one on test day.

Elimination method:

1. Multiply one or both equations so that one variable has matching (or opposite) coefficients.
2. Add or subtract the equations to eliminate that variable.
3. Solve for the remaining variable, then back-substitute to find the other.

Substitution method:

1. Solve one equation for one variable (e.g., $y = ...$ ).
2. Substitute that expression into the other equation.
3. Solve for the single variable, then back-substitute.

Watch for special cases:

• If your variables cancel out and you get a false statement like $0 = 5$, the system has no solution.
• If your variables cancel out and you get a true statement like $0 = 0$, the system has infinitely many solutions.

Keep your work neat. Systems involve a lot of steps, and small arithmetic errors are easy to make.

😅 Tips for Solving Absolute Value Inequalities

The absolute value of a number is its distance from zero on the number line. When solving absolute value inequalities:

1. Isolate the absolute value on one side of the inequality.

2. Split into two cases:

   • For $|A| < k$: rewrite as $-k < A < k$ (a compound inequality).
   • For $|A| > k$: rewrite as $A > k$ or $A < -k$ (two separate inequalities).

3. Solve each case for the variable.

4. Find the overlap. If you have a system of absolute value inequalities, the final answer is the set of values that satisfies all of them at the same time.

🤓 Applying Your Knowledge: Solving Equations, Inequalities, and Systems

The following questions are from the 2020 SAT Study Guide (all credit to College Board).

Solving Linear Equations Practice

What is the solution to the following equation?

$-2(3x - 2.4) = -3(3x - 2.4)$

Here's how to solve it step by step:

1. Expand both sides by distributing: $-6x + 4.8 = -9x + 7.2$

2. Move all x-terms to one side and constants to the other: $-6x + 9x = 7.2 - 4.8$

3. Combine like terms: $3x = 2.4$

4. Divide both sides by 3: $x = 0.8$

You can verify by plugging $x = 0.8$ back in. Left side: $-2(3(0.8) - 2.4) = -2(0) = 0$. Right side: $-3(3(0.8) - 2.4) = -3(0) = 0$. Both sides equal 0, so $x = 0.8$ is correct. No calculator needed.

Systems of Inequality with Absolute Value Practice

Which of the following values of x satisfies both inequalities $|x - 1| < 3$ and $|x + 2| > 4$?

A) x = 3

B) x = -6

C) x = -4

D) x = -3

The answer is A.

Solve each inequality separately:

First inequality: $|x - 1| < 3$

Rewrite as $-3 < x - 1 < 3$. Add 1 to all parts: $-2 < x < 4$.

Second inequality: $|x + 2| > 4$ Split into two cases: $x + 2 > 4$ or $x + 2 < -4$. Subtract 2: $x > 2$ or $x < -6$.

Now find the overlap. You need $x$ to be between $-2$ and $4$ (from the first inequality) AND either greater than 2 or less than $-6$ (from the second). The region $x < -6$ doesn't overlap with $(-2, 4)$, but the region $x > 2$ does. So the combined solution is $2 < x < 4$.

Among the answer choices, only $x = 3$ falls in that range.

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📋 The Relationships among Linear Equations, Lines in the Coordinate Plane, and the Contexts They Describe

Now that you can handle systems algebraically, this section covers what they look like on a graph and how to extract information from the coordinate plane.

🧠 What You Need to Know: Graphing Linear Equations and Systems

Graphing Systems of Equations

To graph a linear equation:

1. Convert to slope-intercept form ($y = mx + b$) if it isn't already. This makes the slope and y-intercept easy to identify.
2. Identify the slope (m) and y-intercept (b).
3. Plot the y-intercept at the point $(0, b)$.
4. Use the slope to find more points. From the y-intercept, rise (move up/down) by the numerator and run (move right) by the denominator of the slope. Draw a straight line through your points.

Once both lines are graphed, the number of solutions is determined by how the lines relate to each other:

• One solution (✅): The lines intersect at exactly one point. That point is the solution.
• No solution (❌): The lines are parallel (same slope, different y-intercepts). They never cross.
• Infinite solutions (♾️): The lines are identical (same slope, same y-intercept). Every point on the line is a solution.

Analyzing Systems of Equations Without Graphing

You don't always need to graph. Once both equations are in slope-intercept form, compare them:

• Same slope, different y-intercepts → parallel lines → no solution.
• Same slope, same y-intercept → identical lines → infinitely many solutions.
• Different slopes → the lines cross → exactly one solution.

Key Formulas

Slope formula (given two points $(x_1, y_1)$ and $(x_2, y_2)$):

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Point-slope form (given a point $(x_1, y_1)$ and slope $m$):

$y - y_1 = m(x - x_1)$

This is useful when you know a point on the line and the slope but need to write the full equation.

Perpendicular lines: If line A has slope $m_a$, then any line perpendicular to it has slope $m_b = -\frac{1}{m_a}$. Perpendicular lines intersect at a right angle.

Graphing Inequalities

1. Graph the boundary line. Treat the inequality as an equation and graph the line normally.
2. Determine the line style. Use a solid line for $\leq$ or $\geq$ (the boundary is included). Use a dashed line for $<$ or $>$ (the boundary is not included).
3. Shade the correct region. For $>$ or $\geq$, shade above the line. For $<$ or $\leq$, shade below the line. If you're unsure, test a point like $(0, 0)$: if it satisfies the inequality, shade the side containing that point.

🤓 Applying Your Knowledge: Graphing Linear Equations and Systems

Solving Systems of Equations Algebraically and Graphically Practice

How many solutions (x, y) does the given system of equations have?

$2y + 6x = 3$

$y + 3x = 2$

A) Zero

B) Exactly one

C) Exactly two

D) Infinitely many

The correct answer is A.

Algebraic approach: Multiply the second equation by 2: $2y + 6x = 4$. Now compare it with the first equation, $2y + 6x = 3$. Subtracting one from the other gives $0 = 1$, which is a false statement. That means no values of x and y can satisfy both equations, so there are zero solutions.

Graphing approach: Rewrite both in slope-intercept form. First equation: $y = -3x + 1.5$. Second equation: $y = -3x + 2$. Both lines have slope $-3$ but different y-intercepts (1.5 vs. 2), so they're parallel and never intersect.

Writing the Equation of a Line: Point-Slope Form Practice

Which of the following is the correct equation of a line that passes through the point (2, 3) and has a slope of 2?

A) y = 2x - 1

B) y = 2x + 1

C) y = -2x + 5

D) y = -2x - 5

The correct answer is A.

1. Start with point-slope form: $y - y_1 = m(x - x_1)$

2. Substitute the point $(2, 3)$ and slope $m = 2$: $y - 3 = 2(x - 2)$

3. Distribute: $y - 3 = 2x - 4$

4. Add 3 to both sides: $y = 2x - 1$

Quick check: plug in $x = 2$. You get $y = 2(2) - 1 = 3$. That matches the given point, so the equation is correct.

Graphing Inequalities Practice

How would you graph the solution to the inequality $x + 2y < 4$?

1. Graph the boundary line $x + 2y = 4$. Find two points:

   • Set $x = 0$: $2y = 4$, so $y = 2$. Point: $(0, 2)$.
   • Set $y = 0$: $x = 4$. Point: $(4, 0)$.
   • Plot both points and draw a dashed line through them (dashed because the inequality is strict $<$, not $\leq$).

2. Determine which side to shade. Test the point $(0, 0)$: $0 + 2(0) = 0 < 4$. This is true, so shade the side of the line that contains $(0, 0)$, which is below the line.

3. Verify. Pick a point from the shaded region (like $(1, 1)$): $1 + 2(1) = 3 < 4$. True. Pick a point from the unshaded region (like $(3, 2)$): $3 + 2(2) = 7 < 4$. False. The shading is correct.

The final graph is a dashed line through $(0, 2)$ and $(4, 0)$ with the region below it shaded.

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⭐️ Closing

That covers all four skill areas within Heart of Algebra. Since this topic makes up a third of the SAT Math section, getting comfortable with these skills will have a big impact on your score. Practice translating word problems into equations, solving systems both algebraically and graphically, and always double-check your work.

For more resources, check out SAT Math Section Tips and Tricks and "What are the SAT Math Test Questions Like?"