SAT: Heart Of Algebra

The Collegeboard divides the SAT Math section into 4 categories: Heart of Algebra, Problem Solving and Data Analysis, Passport to Advanced Math, and Additional Topics in Math. Of these, Heart of Algebra accounts for the largest portion on the SAT Math section - approximately 33% or 19 questions. There are 8 questions on Section 3 (No Calculator Section) and 11 on Section 4 (Calculator Section). So, you must be very well prepared for this part of the math section.

What Is Inside The Heart Of Algebra?

• Linear Equations
• Linear Inequalities
• System Of Equations
• Absolute Value
• Lines In The Coordinate Plane

Linear Equations

You might be asked to solve a given linear equation or create a linear equation from the word problem context provided. These questions will definitely ask you to define one or more variables that represent quantities in the question or write equation(s) to represent the relation between quantities given in the question. Your job is always to interpret the solution in terms of what the question is asking. Be sure to check the question’s context and the meaning of the variables before picking the final answer. On difficult questions, it might happen that your answer doesn’t match any of the answer choices. In such cases, consider expanding, simplifying, or rearranging your solution. 

Linear Inequalities

You might be asked to solve a provided linear inequation or create a linear inequation from the word problem context provided. These questions are usually among the most challenging on the SAT. These questions will generally ask you to solve an inequality or an inequality graph or find an alternate inequality equation that fulfills the given conditions. 

System Of Equations

These questions will require you to deal with more than one variable and create more than one equation or inequality. The simpler questions will ask you to solve for one variable when the information for more than one variable is provided. The harder questions will require you to create equations and solve for one or both the variables. 

Absolute Value

Absolute value questions are generally straightforward. However, you definitely need to know the rules of absolute values to solve these questions correctly. Be sure to remember that absolute value is always more than equal to zero. So, |x| ≥ 0.

Absolute value is the distance between two points on the number line. This understanding helps clarify why the absolute value is always more than equal to zero. This also means that an absolute value equation will always have two solutions. For example, if |x - 1| = 2, then x = 3 or -1. 

Lines In The Coordinate Plane

A line is a linear equation but on a graph. These questions need you to know the slope of a line, the y-intercept, and the x-intercept. It will be helpful to know the relation between two parallel lines and perpendicular lines. So, know your equation y = mx + c very well. Here, m is the slope and c is the y-intercept. 

Strategies To Score High On Heart Of Algebra

Strategy 1: Plugging in

These questions have variables in the question and also in the answer choices. The trick is to replace the variable with a number and solve the question.

Q1. 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 that Miguel charges if it takes him x hours to edit a manuscript, where x > 2?

1. C = 20x

2. C = 20x + 10

3. C = 20x + 50

4. C = 20x + 90

Solution:

Plugging in means replacing the variable with a number.

Let x = 3 (as x > 2).

Miguel charges $50 for the first 2 hours.

He charges $20 for the next 1 hour.

So, he charges a total of $50 + $20 = $70.

Put x = 3 in the answer choices and check which answer matches $70.

1. C = 20x = 20(3) = $60

2. C = 20x + 10 = 20(3) + 10 = $70

3. C = 20x + 50 = 20(3) + 50 = $110

4. C = 20x + 90 = 20(3) + 90 = $150

So, the answer is (B).

Q2. Adam is x years old and 4 years older than Bob. How old was Bob 7 years ago?

1. x - 3

2. x - 4

3. x - 7

4. x - 11

Solution:

Let x = 10.

So, Adam’s age is 10.

He is 4 years older than Bob, so Bob’s age at present = 10 - 4 = 6 years.

Hence, 7 years ago, Bob was 6 - 7 = -1 year. 

Replace x with 10 and check which answer choice gives -1.

1. x - 3 = 10 - 3 = 7

2. x - 4 = 10 - 4 = 6

3. x - 7 = 10 - 7 = 3

4. x - 11 = 10 - 11 = -1

Only answer choice (D) works. 

Strategy 2: Hidden Plugin

These questions do not have a variable in either the question or the answer choices. But, these questions have answer choices as fractions or percentages. 

Q3. Adam gave 30% of his money to his friend Bob. He then gave 60% of his remaining money to his friend Cindy. What percentage of his original amount is he left with after he gave the money to Cindy?

1. 10%

2. 18%

3. 28% 

4. 36%

Solution:

Let the original amount that Adam has equal $100. (As the question deals with percentages, 100 makes it easy to make calculations)

He gives 30% to Bob. So, now Adam has 100 - 30 = $70.

He gives 60% of this amount to Cindy. So he has 70 - 42 = $28.

So, he has (28/100) * 100 = 28% of his original amount.

The answer is (C).

Strategy 3: Plugging in the answer 

These questions have answer choices as integers - positive or negative.

Q4. A store sells pens for $7.50 each and pencils for $5.00 each. The store earns $1822.50 in one day from selling a total of 307 pens and pencils. How many pens were sold that day?

1. 37

2. 89

3. 115

4. 202

Solution:

|   | Pens | Earnings From

Pens 

| Pencils |

Earnings From

Pencils

| Total Earnings | | --- | --- | --- | --- | --- | --- | | A) | 37 | 277.5 | 270 | 1350 | 1627.5 | | B) | 89 | 667.5 | 218 | 1090 | 1757.5 | | C) | 115 | 862.5 | 192 | 960 | 1822.5 | | D) | 202 | 1515 | 105 | 525 | 2040 |

Only answer choice (C) works. 

Now you know what to expect on the SAT for Algebra, but make sure that you prepare for the other topics as well. 

Happy prepping!!