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Heart of Algebra on the SAT Test

Heart of Algebra Subscore on the SAT Test

Included topics: linear functions; linear equations; systems of linear equations; linear inequalities

Heart of algebra subscore is one of the three SAT math test subscores.

Heart of algebra subscore includes four core algebra topics: linear functions, linear equations, systems of linear equations and linear inequalitiesEach one of these 4 topics is explained on a different page on this site, see the left menu or the top menu (under heart of algebra).

The skills tested in heart of algebra subscore:

  • The skills tested in heart of algebra subscore include creating, analyzing and solving equations and inequalities.
  • Some questions test your ability to represent a word problem algebraically. This will require defining variables, writing expressions, solving equations and interpreting the solution in terms of the question.
  • Other questions focus on interpreting the relationship between graphic and algebraic presentations.

About the test:

  • Heart of algebra subscore questions include multiple choice questions and student produced response questions.
  • The use of a calculator is permitted for part of the questions.
  • The grade of heart of algebra subscore is reported on a scale of 1 to 15.

Linear functions SAT topic

A linear function is an equation that represents a relationship between two variables, most commonly called x and y. It is called linear because it can be graphed as a straight line in the xy-plane. 

Creating a linear function requires writing a function that describes a given word problem without solving it.

Finding the value of a function requires solving a function given the values of its variables.

Finding the input that corresponds to a given output requires to calculate the input of a function given the value of the output.

Interpretation of a linear function includes analyzing a linear function in its slope-intercept form: y=mx+b (rewriting the function from its standard form ax+ty=c, calculating the slope and the intercept of the function and analyzing of the graph of the function).

Finding the equation of a linear function requires to write the equation of a linear function y=mx+b given the values of its points or its slope b.

Graphic presentation of a linear function requires identifying a graph of a given linear function or finding the equation of a linear function given its graph.

Analyzing relationships between two linear functions includes finding the slope of a linear function given the slope of another function or finding the equation of a linear function given the equation of another function.

The linear functions formula sheet is given below, it includes the following formulas:

Two forms in which we can write a linear function: the standard form ax+ty=c and the slope intercept form y=mx+b

The formula for calculating the value of the slope of a linear function given 2 points a and b: m=(ya-yb)/(xa-xb).

Two formulas the represent the relationships between two linear functions: parallel lines slopes formula m1=m2 and perpendicular lines slopes formula m1*m2=-1.

Note that all the formulas are explained in detail on this page. 

Linear functions SAT formula sheet

Linear equations SAT topic

A linear equation is an algebraic equation in which each term has an exponent of one. It is called linear because it can be graphed as a straight line in the xy-plane. Since this is a fundamental topic it appears in many SAT questions. 

Calculating an output of an expression requires calculating the value of an expression. In some questions we are given a word problem and in other questions we are given an equation that we must use it order to solve the expression.

Solving linear equations- In these questions we are given an equation and we are asked to solve it. Some questions may include fractions or absolute values. 

Linear equations that don’t have one solution- In these questions we are asked to solve an equation that has no solution or an infinite number of solutions.

Creating and solving an equation from a word problem- In these questions we are given a word problem from which we need to write an equation and solve it.

The linear equations formula sheet is given below, it includes 3 types of basic formulas:

Calculations with negative numbers formulas– these are 5 very basic formulas that include multiplying and dividing positive and negative numbers.

Distributing and combining like terms formulas-one formula for opening parentheses (distributing) and one formula for creating them (combining). 

Calculation with fraction formulas– includes three basic formulas for addition/ subtraction, multiplication and division of fractions.

Note that all the formulas are explained in detail on this page. 

Linear equations formula sheet

Systems of linear equations SAT topic

A system of linear equations is a set of two or more linear equations. In SAT questions we usually see systems of equations containing 2 equations with 2 variables x and y.

If we are given a word problem, we first need to define the variables and then write 2 equations with these variables.

Creating a system of linear equations- In these questions we need to create a system of linear equations from a word problem.

Solving a system of linear equations- In these questions we need to solve a system of linear equations. In order to solve the equations, we need to reduce 2 equations with 2 variables to 1 equation with 1 variable. This should be done with either substitution or elimination.

Determining the number of solutions for a system of linear equations- In these questions we are asked to determine if the system of equations has one solution, no solution and an infinite number of solutions.

Graphic presentation of a system of linear equations- In these questions we need to identify a graph of a given system of equations.

Linear inequalities SAT topic

A linear inequality is linear equation in which the equal sign is replaced by one of the symbols of inequality. The solution of a linear equation is a range of values, rather than one specific value.

A system of linear inequalities is like a system of linear equations, the difference is in the sign. We use the system of inequalities when we a have a word problem with number of constrains instead of one (we write an inequality for each constrain). 

Creating and solving a linear inequality requires creating a linear inequality from a word problem and solving it. Note that solving a linear inequality may require changing the direction of the inequality sign.

Finding values that don’t satisfy an inequality– In these questions we need to find values that are not a solution for a linear inequality.

Graphic presentation of a linear inequality requires identifying a graphic presentation of an inequality. The answers to the inequality are represented by the area below or above the line of its equation. 

Creating a system of linear inequalities- In these questions we need to create a system of linear inequalities from a word problem.

Finding possible solutions for a system of linear inequalities– In these questions we need to decide which values satisfy a given system of linear inequalities. Sometimes we will need to translate a word problem into a system of inequalities as explained before.

Solving a system of linear inequalities requires turning a system of linear inequalities to one inequality that can be solved easily. This can be achieved using additional data or by turning one of the inequalities to an equation. We may also need to translate a word problem to a system of inequalities.

Graphic presentation of a system of linear inequalities– In these questions we need to identify a graphic presentation of a system of inequalities given its formulas or to write the formulas of the system that correspond to a given graphic presentation. 

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