Math Study Guide for the SAT Exam

Linear Equations and Functions

Understanding linear equations and functions is a critical part of the SAT Math section. A linear equation shows the relationship between variables whose highest power is \(1\). A linear equation does not need to include more than one variable. When one changes, the other changes by a steady rate. A linear function is a linear equation that is also a function, meaning that when it is graphed on a coordinate plane, no \(x\)-value maps to more than one \(y\)-value. The term “linear” comes from the fact that when these equations are graphed in the coordinate plane, they form a straight line. The sections below will outline how to solve linear equations in one and two variables, how to interpret and graph them, and how systems of equations and inequalities extend these ideas.

Linear Equations in One Variable

A linear equation is one that forms a straight line when plotted on a graph. Many of the linear equations you will see on the SAT will have just one variable.

Consider the following single-variable equation:

\[2 - 3(x + 2) - 4x = 3 + 4x\]

We need to solve this equation by manipulating it such that a single \(x\) is on one side of the equation and a single value is on the other side. In this case, we will begin by distributing the \(-3\) through the parentheses:

\[2 -3x - 6 - 4x = 3 + 4x\]

Now, we’ll combine like terms on the left side of the equation:

\[-4 - 7x = 3 + 4x\]

There are multiple ways to continue. We can either add \(7x\) to both sides of the equation, subtract \(4x\) from both sides of the equation, add \(4\) to both sides of the equation, or subtract \(3\) from both sides of the equation. To avoid a negative sign in the variable’s coefficient, let’s add \(7x\) to both sides:

\[-4 = 3 + 11x\]

Because we want all of the numbers on one side and only one \(x\) on the other side, we subtract \(3\) from both sides. Remember that any operation performed on one side of the equation must also be performed on the other side of the equation to maintain equality:

\[-7 = 11x\]

Notice that the variable is currently multiplied by \(11\). To undo this operation, and to isolate just one \(x\), it is necessary to divide the right side by \(11\). To keep the equation equal, we must also divide the left side by \(11\):

\[\frac{-7}{11} = \frac{11x}{11}\] \[\frac{-7}{11} = x\]

Our answer can be verified by substituting \(\frac{-7}{11}\) into our original \(x\) and checking to see if the equation is true:

\[2 - 3(\frac{-7}{11} + 2) - 4 \cdot \frac{-7}{11} = 3 + 4 \cdot \frac{-7}{11}\] \[2 - 3(\frac{15}{11}) + \frac{28}{11} = 3 - \frac{28}{11}\] \[2 - \frac{45}{11} + \frac{28}{11} = 3 - \frac{28}{11}\] \[\frac{22}{11} - \frac{45}{11} + \frac{28}{11} = \frac{33}{11} - \frac{28}{11}\] \[\frac{5}{11} = \frac{5}{11}\]

The solution is correct.

On exams, presuming enough time is available, always verify your answer using the method above. We won’t take the time to do so with every problem in this guide, but if it helps you, we recommend always verifying the answers while you practice until you feel completely comfortable with the concepts.

Linear Equations in Two Variables

A linear equation containing one variable can be numerically solved for the variable. A linear equation containing two (or more) variables can only be solved in terms of those variables. For example:

\[2y + 6x = 10\]

We can solve this equation for \(x\) or \(y\), but the other side of the equation will still contain the other variable. To solve for both variables, we would need to have two distinct equations, known as a system. For now, consider the following equation in two variables:

\[3xy + 4x - 2y + 3(x - y) = 4y + 2x\]

We can either solve in terms of \(x\) or \(y\). Let’s solve for \(y\) in terms of \(x\). To do so, we will end up with an equation with \(y\) on one side and everything else on the other.

The first step entails distributing the \(3\) through the parentheses so that every term is distinct:

\[3xy + 4x - 2y + 3x - 3y = 4y + 2x\]

Now, move every term containing a \(y\) to one side of the equation and everything else to the other side of the equation:

\[3xy - 2y - 3y - 4y = 2x - 4x - 3x\]

Next, combine like terms:

\[3xy - 9y = - 5x\]

Here, the \(y\) can be factored out of the expression on the left, meaning we remove the common factor out of the terms (essentially the opposite of distributing):

\[y(3x - 9) = -5x\]

We can now divide both sides by the parentheses to solve for \(y\) in terms of \(x\):

\[y = \frac{-5x}{3x-9}\]

It might not look very clean, but that is our answer.

Linear Functions

In the real world, we often determine the relationship between two quantities, such as how far a vehicle travels over a certain amount of time or the effect of online ratings on a restaurant’s business. With these relationships, we can use time to calculate the distance a car traveled or average star ratings to predict the restaurant’s expected profit.

Because we’re free to plug in whatever value we want for time or rating, we call these quantities variables. Furthermore, we use the term function to describe relationships like these, ones that associate one value to each value the input variable could have. As you will see, the input doesn’t have to be just a variable. It can be an expression, so we more generally refer to a function’s input as its argument.

Function Basics

If the function is \(f\) and its argument is \(x\), we write the function as \(f(x)\) to make it clear that the value of \(f\) depends on the value of \(x\). For example, if we wanted \(f\) to be a function that doubles and adds one, we would write:

\[f (x) = 2x + 1\]

We could just as well write:

\[f (a) = 2a + 1\]

The variable in the argument is just a placeholder. In fact, we could even insert another expression in its place, such as:

\[f(b - 3) = 2(b - 3) + 1\]

The notation just means, “When you replace the placeholder variable in the parentheses with something, replace it with the same something on the other side of the equation.”

Graphing Linear Functions

A linear function is a function of one variable that graphs as a line. The input value is commonly called \(x\), and the output value, \(f(x)\), is commonly called \(y\). Functions of the form \(f(x) =\) can be graphed in the coordinate plane by selecting values for \(x\), determining their corresponding \(y\) values, and then placing points at each of the calculated \((x,y)\) values.

The nature of the function determines whether a straight or a curved line is drawn to connect each point. Consider the following linear function:

\[f(x) = 3x - 2\]

To graph this function on a coordinate plane, input two distinct \(x\) values and find their corresponding \(f(x)\) values. Plot both points on the coordinate plane, then draw a straight line through them. For example, let’s select \(x=0\) and \(x = 2\). Thus, we get:

\[f(0) = 3(0) - 2 = -2 \text{ and }f(2) = 3(2) - 2 = 4\]

The coordinate points are \((0,-2)\) and \((2,4)\). Below, we show the graph:

Familiarity with the common equations for a line sometimes simplifies the graphing process. You should be comfortable using the following:

• the slope-intercept form—\(y = mx + b\)

• the point-slope form—\((y - y_1) = m(x - x_1)\)

• the standard form—\(Ax + By = C\)

where \(m\) is the slope, \(b\) is the \(y\)-intercept, \((x_1, y_1)\) is a point on the line, and \(A\), \(B\), and \(C\) are constants.

Systems of Two Linear Equations in Two Variables

As discussed earlier, when we have a linear equation in two variables, we can only solve in terms of one or the other variable. However, when we have multiple equations relating the same variables, we call that a system of equations, and we can solve for both variables. Solving for the variables means we must find values for each variable for which every equation in the system holds true. This is known as the solution set.

Consider the following system of equations:

\[\begin{Bmatrix} 4x + y = 6\\ -3x - 2y = 8 \end{Bmatrix}\]

There are three methods for solving this system available to you: elimination, substitution, and graphing.

Elimination

To solve a system by elimination requires multiplying one of the equations by a value such that combining both equations will result in one of the variables being canceled out. We’ll use the above system to illustrate this technique.

Consider what it would take to eliminate one of the two variables. In this case, by multiplying the top equation by \(2\), the coefficients of the \(y\) variable will cancel out and the \(x\) variable can be found:

\[2(4x + y = 6)\]

becomes

\[8x + 2y = 12\]

Adding the first equation to the second yields:

\[\begin{array}{r} 8x + 2y = 12&\ \underline{\quad-3x - 2y = 8}&\\ \end{array}\] \[5x = 20\] \[x = 4\]

Now that \(x\) has been solved, its value can be substituted into either equation to solve for \(y\):

\[4(4) + y = 6\] \[16 + y = 6\] \[y = -10\]

The solution set is \((4, -10)\).

Note: To double-check our result, we can plug our answers into the second equation:

\[-3(4) - 2(-10) = -12 + 20 = 8\]

That’s correct, so we know our answer is right.

Substitution

Solving by elimination is not always the most efficient method. Sometimes it is best to solve a system using substitution. The goal of solving by substitution is to express one of the variables in terms of the other. The expression can then be substituted into the other equation, and the value for the variable can be found. Let’s try this with our original system of equations:

\[\begin{Bmatrix} 4x + y = 6\\ -3x - 2y = 8 \end{Bmatrix}\]

The first step is solving the first equation for \(y\):

\[y = -4x + 6\]

Now, we can substitute this expression for \(y\) into the other equation:

\[-3x - 2(-4x + 6) = 8\] \[-3x + 8x - 12 = 8\] \[5x = 20\] \[x = 4\]

Substituting this value of \(x\) into either equation will yield the value of \(y\), \(-10\), just as we got using the elimination method.

Graphing

Solving a system of equations using the graphing method simply entails graphing both lines and finding their point of intersection. If we were to graph the two equations from our example system, the two lines would intersect at the point \((4, -10)\).

Not every system of equations will have lines that intersect, however. There are three cases of solutions:

• The two lines do not intersect.—This happens when the two lines are parallel but have different \(y\)- or \(x\)-intercepts. In this case, there is no solution.

• The two lines intersect at one point.—This happens when the two lines are distinct but do not share the same slope. In this case, there is one solution.

• The two lines intersect at every point.—This happens when the two lines are the same. In this case, there are infinite solutions.