Page 1 - 11th Grade Mathematics: Algebra Study Guide for the SBAC

How to Prepare for the Algebra Questions on the SBAC Mathematics Test

General Information

The skills you have learned in algebra are used in a variety of mathematical situations and you will find that they are especially helpful in problem-solving, which is a big part of this test. It is important not only to be able to do the operations but to be able to explain what you did and tell why.

Skills to Understand and Be Able to Use

Expressions

An expression is a combination of variables, regular numbers, and operations such as addition, division, subtraction, and multiplication. An example is \(3x - 5y + 4\)

Structure

When interpreting expressions, often you can solve a problem by considering the structure of the expression rather than using algebra to simplify or manipulate it. For example, it might be enough to know if it might evaluate to be positive, negative, or zero. An expression with a set of multiplied terms will be zero if any of the terms is zero. Another common structure is to match factoring patterns, such as noticing whether an expression is a difference of squares.

Interpret in Terms of Context

The context of a problem is important in considering solutions. For example, if the problem involves physical items, you cannot have a negative number of items. This knowledge may guide you while analyzing a problem. In this case, if an expression will always have negative values, it cannot be a valid description of a number of items.

Parts of an Expression

In our sample expression \(3x - 5y + 4\), there are three terms, \(3x\), \(5y\), and \(4\). Terms are quantities that are added or subtracted. In the term \(3x\), \(3\) and \(x\) are factors, because they are multiplied together. A coefficient is a factor that is an ordinary number. So \(3\) is the coefficient in the term \(3x\).

Complicated Expressions

You can interpret complicated expressions by considering terms or factors as a single entity. For example, the expression \(C(3 + x)t\) is the product of \(C\) and two other factors that don’t depend on \(C\). We can tell that the expression will be zero if \(x = -3\), or it might be negative if \(x /lt -3\).

Equivalent Forms

Expressions are equivalent if we can apply valid algebraic changes to one of them and make it look like the other one. \(5(2y - 2)\) and \(10y - 10\) are equivalent because we can use the distributive rule on the first one and it will be written the same as the second one. Also, if you substitute the same value for the same variables in each expression, they are equivalent if they give the same result. Often, by factoring an expression into an equivalent expression, you can better understand whatever quantity the expression represents.

Finite Geometric Series

A geometric series is a series of terms where each term there is a constant ratio between the terms. Each term can be found by multiplying the previous term by the common ratio. Finite means that it ends after a specific number of terms. For example, \(1,\; 0.5, \;0.25, \;0.125\) is a geometric series where each term is multiplied by \(0.5\) to get the next term. Although it looks complicated, the formula for the sum is easy to derive using some simple algebra, and you should be aware of how to derive it.

Mortgage Payments

Mortgages are made by compound interest, where the same interest rate is applied to the remaining loan amount each month. This is an application of a geometric series and you can use the formula for the sum of the series to solve problems involving compound interest payments.

Rational Expressions

A rational expression is a fraction where both the numerator and denominator have expressions. Since denominators cannot be zero, any value of a variable that makes the denominator evaluate to zero is not allowed. Rational expressions are often simplified by factoring. It is important to recognize factoring patterns such as the difference of squares in the structure of these expressions.

Equations and Inequalities

Equations and inequalities state a relationship between two expressions. For an equation, the expressions are equal. An inequality specifies which expression has a greater or lesser value.

Creating Equations and Inequalities

To create an equation or inequality, you will first need to assess and interpret the information you are given. You write expressions based on the information and then look for information that tells you the relationship between these expressions. In some cases, one of the expressions may simply be a number.

Equations and Inequalities in One Variable

Relationships involving just one quantity are often written as equations or inequalities with just one variable. Simply writing the equation is often not enough, you must also solve it in order to solve the original problem. You should know how to solve equations that arise from linear, quadratic, rational, and exponential functions.

Equations in Two or More Variables

When more than one quantity exists in a problem, you will need equations in two or more variables. In some cases you can solve a system in two variables by graphing the equations on coordinate axes, choosing the appropriate labels and scales.

Representing Constraints

When modeling or representing real-world problems, there are often solutions to the equations that are not viable solutions in real life. For example, food cannot have negative amounts, costs, or nutritional values. These constraints may be represented as inequalities such as \(x \ge 0\).

Rearranging Formulas

A formula may have several variables, such as Ohm’s law \(V = IR\). Problems using the formula will give you information to determine all but one of them. The remaining variable is not always on the left side of the formula. In this case you must rearrange the formula, as if you were solving an equation, to highlight the variable you need to solve for.

Solving Equations

Solving equations is a process of reasoning, such as “If I add the same amount to both sides I don’t change the result.” On this test, you may be called on to explain the reasoning, or to explain errors in reasoning, instead of (or in addition to) actually solving the equation.

Mastering Linear Equations

The linear equation is a powerful tool in problem-solving and especially modelling real-world situations. It is important to master this tool, recognizing or rearranging to get the form \(y = mx + b\), graphing the result, and interpreting what the slope and intercept represent in a specific problem.

Simple Radical and Rational

While solving simple radical and rational equations, extraneous solutions may be introduced in the process, For example, rational equations have disallowed values that make the denominator evaluate to zero. The process of simplifying may appear to remove one of these values. For example, \(\dfrac{5x}{x^{2} + 3x}\) simplifies to \(\dfrac{5x}{x(x + 3)} = \dfrac{5}{x + 3}\). It would seem that \(x=0\) is an allowed solution because it does not make the simplified denominator evaluate to zero. But it does make the original denominator evaluate to zero, so it is an extraneous solution that is not allowed.

Explaining Each Step

You will be asked to explain steps in solving equations in addition to, or even instead of, actually solving them. Generally, the justification is that you are performing the same operation on equal quantities, thus keeping them equal. The reason for choosing a particular operation is to simplify until you have a statement that the variable is equal to a certain quantity.

Letter Coefficients

Some equations, such as \(y = mx + b\), look like they have several variables. In this case, \(m\) and \(b\) are not variables, they are parameters that are known and constant for any specific function. Usually, you know values for \(m\), \(b\), and one of the variables, or you can find them from information given in the problem. Sometimes, however, you may be given values for both variables and one parameter and you will solve for the other parameter as if it were a variable.

Quadratic Equations

Quadratic equations in one variable are solved in several ways, often involving the recognition of factoring patterns. When methods using these patterns don’t work, you can use the quadratic formula.

Completing the Square— Completing the square works by fitting a quadratic equation to a form where it can be factored to \((x-p)^{2}\) or \((x+p)^{2}\). Suppose you have: \(x^{2} - 4x + 2 = 0\).
You notice that if it was \(x^{2} - 4x + 4 = 0\), you could factor it to \((x - 2)^{2} = 0\).
No problem, just add 2 to both sides if the original: \(x^{2} - 4x + 2 +2 = 0 +2\).

Now simplify to \(x^{2} - 4x + 4 = 2\),
and factor to \((x - 2)^{2} = 2\). Just take the square root of both sides and you have a solution.

Note that you can always find a number to add or subtract, but it will often be a fraction.
To find the number, take the middle coefficient, divide it by \(2\) and then square it. In this case: \(4 \div 2 = 2 \rightarrow 2^2 = 4\).

Also note that this is how the quadratic formula is derived from \(ax^{2} + bx +c = 0\).

Inspection— Solving by inspection means to recognize that the equation fits a simple factoring pattern and use it. Otherwise you have to complete the square (if it almost fits a pattern) or use the quadratic formula. Remember that if you get complex numbers as solutions that they are written as \(a \pm bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary number \(\sqrt{-1}\).

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