Heart of Algebra

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We have lots of free resources and videos to help you prepare for the SAT. These materials are for the redesigned SAT which is for you if you are taking the SAT in March 2016 and beyond.

Heart of Algebra consists of questions that will test your ability to analyze and and solve linear equations and systems of linear equations. You need to be able to create linear equations and inequalities to represent relationships between quantities and to solve problems. You need to understand and use the relationship between linear equations and inequalities and their graphs to solve problems.

Many Heart of Algebra questions can be answered using the following steps:

  1. Define the variables to represent the quantities in the question.
  2. Write the equations, expressions, inequalities, or functions to represent the relationships described in the question.
  3. Solve the equations or inequalities and interpret the solution in terms of what is required in the question.
  4. You may need to expand, simplify or rearrange your solution to match an answer choice.

There are many ways that you can be tested and practicing different types of questions will help you to be prepared for the SAT.

The following video lessons will show you how to solve a variety of Heart of Algebra questions in different situations.

Linear Equations, Linear Inequalities and Linear Functions
Solving linear equations & inequalities (Basic)
Solving linear equations & inequalities (Harder)
Interpreting linear functions (Basic)
Interpreting linear functions (Harder)
Linear equation word problems (Basic)
Linear equation word problems (Harder)
Linear inequality word problems (Basic)
Linear inequality word problems (Harder)
Linear function word problems (Basic)
Linear function word problems (Harder)

Solve Linear Equations 1
Solve Linear Equations 2
Linear equation word problems 1
Linear equation word problems 2
Linear function word problems
Solve Linear Inequalities 1
Solve Linear Inequalities 2
Solve Linear Inequalities 2

Line Translation

Absolute Value
Heart of Algebra includes absolute value expressions, inequalities, and equations.
The absolute value of any real number is nonnegative. Therefore, |-x| = |x|, for any real number x.
For any real number a and b, |a - b| is the distance between a and b on the number line.
You will need to know how to solve absolute value inequalities and to find the absolute value inequality when given the interval or range of values.

Absolute Value Inequality
Describe Intervals Using Absolute Values
Absolute Inequality Problem
Absolute Value Expression
Solve Absolute Value Inequality

Systems of Linear Equations and Inequalities
You can use either the elimination or substitution method when solving a system of linear equations. Look at the equations closely because sometimes one method may get you the answer quicker that the other.
You will need to know the conditions when a systems of linear equations has no solution, one solution, and infinite solutions.

Solving systems of linear equations (Basic)
Solving systems of linear equations (Harder)
Systems of linear equations word problems
Systems of linear inequalities (Basic)
Systems of linear inequalities (Harder)

Solve systems of linear equations 1
Solve systems of linear equations 2
Systems of linear equations word problems 1
Systems of linear equations word problems 2

Lines in the Coordinate Plane
A system of two linear equations in two variables can be solved by graphing the lines in the coordinate plane. The point of inter section gives the solution to the system.
There are three possibilities:

  1. The lines intersect at one point. Therefore, the system has a unique solution.
  2. The lines are parallel and do not meet. Therefore, the system has no solution.
  3. The lines are identical. Therefore, the system has infinitely many solutions.

If the slopes of the line l and line k are defined (i.e. neither line is a vertical line) then

  1. line l and line k are parallel if and only if they have the same slope.
  2. line l and line k are perpendicular if and only if the product of their slopes is -1.

Graphing Linear Equations (Basic)
Graphing Linear Equations 1
Graphing Linear Equations 2
Graphing Linear Equations 3
Graphing Linear Equations 4
Graphing Linear Equations 5

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.
Mathway Calculator Widget

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