Systems of Equations

A system of equations has more than one equation with more than one variable. The same variables are used in all of the equations. To have a solution, the number of equations must match the number of variables. Usually you will see two equations with two variables.

Solving by Elimination

Some systems are easily solved by deriving an equivalent equation to solve for one variable, then substitute back into one of the original equations to solve for the other one. For the equations:

\[x +y = 8\]

and

\[x - y = -2\]

you can add the equations to get \(2x = 6\), and \(x = 3\). Then substitute in the first equation to get \(y = 5\).

You can prove that this method works because you are adding or subtracting equal quantities to both sides. Multiplication and division also work. For equations such as

\[x + y = 6\]

and

\[3x - 2y = 3\]

multiply the first equation (both sides) by \(2\):

\[2x + 2y = 12\] \[3x - 2y = 3\]

and then add the equations together to get \(5x = 15\) and \(x = 3\).

Substitute in the first equation to get \(y = 3\).

Solving by Substitution

Systems are also solved by substitution.
From one equation, derive an expression for one variable in terms of the other, then substitute this into the second equation.
Substitute the value you find back into the first equation to get the other variable. For example:

\(x + y = 6\) and \(3x - 2y = 3\), solve the first equation for \(x\):

\(x = 6 - y\), and then substitute into the second equation and solve for \(y\):

\[3(6 - y) - 2y = 3\] \[18 - 3y - 2y = 3\] \[18 - 5y = 3\] \[-5y = -15\] \[y = 3\]

Substituting \(3\) for \(y\) back into the first equation gives \(x = 3\).

Solving Approximately

A solution to a system of two linear equations must work in both equations. If the equations are graphed, the solution will be at the intersection of the lines. If the intersection does not fall at exact \(x\) and \(y\) values on the grid, you will have to give approximate values. Be sure to be familiar with graphing lines, such as the slope-intercept form.

Mixed Equation Types

A mixed system with a linear and a quadratic equation can be solved graphically or by substitution. To graph the quadratic equation you will need to know standard forms that are easy to graph, for example finding the vertex of a parabola or the center and radius of a circle. A graph may only provide an approximate solution if the intersections do not fall on the grid.

Matrix Equation and Vector Variables

A linear system of equations can be represented in matrix form. The system
\(x + y = 6\)

and

\[3x - 2y = 3\]

is represented as follows:

\[\begin{bmatrix} 1 & 1 \\ 3 & 2 \\ \end{bmatrix} \times \begin{bmatrix} x \\ y \\ \end{bmatrix} = \begin{bmatrix} 6 \\ 3 \\ \end{bmatrix}\]

Matrices with just one row or one column are called vectors. The general form of a linear system with matrices is \(Ax = b\).

Find an Inverse

For a system represented in matrix form \(Ax = b\), the solution is \(x = A^{-1}b\), where \(A^{-1}\) is the inverse of the matrix \(A\). Finding the inverse of a 2 x 2 matrix is not too complicated, but finding the inverse of a 3x3 matrix likely requires technology.

Finding the inverse of a matrix requires calculating a quantity called the determinant of the matrix. If the determinant is zero, there is no inverse for the matrix.

Graphic Representation

Linear functions and many exponential functions can be represented as graphs in the coordinate plane. Generally, these functions have certain forms (such as \(y = mx + b\) for lines) that aid in plotting their graphs without much calculation. Also, you can use technology such as graphing calculators or approximations from tables to make the graph.

Equations in Two Variables

The graph of an equation shows all of the pairs of values (usually \(x\) and \(y\)) that are solutions to the equation. The graph is a line when both variables have an exponent of \(1\) or some form of curve if one or both variables have an exponent greater than \(1\).

Explaining Solutions

If you plot two functions, they might intersect in one or more places. Since each point on the plot represents a solution to the function, and the intersection points are on both graphs, the coordinates of the intersection represent values that solve both functions.

Graphing Inequalities

The graph of a solution to an inequality is an area of the coordinate plane. This area is shaded either above or below the graph of the line formed by making the function an equality. If the inequality is \(\le\) or \(\ge\), the plotted line is solid. For \(\lt\) or \(\gt\), the line is shown as dotted or dashed because the points on the line are not part of the solution. The line shows the solution for \(=\).

Polynomials

A polynomial is a collection of terms containing a variable raised to a power that is not negative. For example, \(5x^{2} - 3x +7\) is a polynomial with three terms, \(5x^{2}\), \(-3x\), and \(7\). The terms can be combined by addition, subtraction, multiplication, or division, but division by a variable (a negative power of a variable) is not allowed.

Operations

You can add, subtract, multiply, and divide polynomials. For addition and subtraction, terms can be combined only if they have the same variables raised to the same powers. These are called like terms. For multiplication, each term in a polynomial must be multiplied by every term in the other polynomial. The well-known FOIL method is a form of polynomial multiplication. For division, the technique of polynomial long division must be used if the divisor polynomial has more than one term. In the case of division, the result may not be a polynomial if it contains a variable raised to a negative power.

Zeros and Factors

A zero of a polynomial is a value of the variable that makes the value of the polynomial equal to zero. It is a solution to the equation you get when you set the polynomial equal to zero and solve. If a polynomial is factored, it is equal to zero if any of the factors equals zero. Quadratic polynomials ( greatest value of the exponent is \(2\)) and some cubic polynomials (greatest exponent is \(3\)) can be factored using common factoring patterns. In this case, you can find the zeros by determining the values that make each factor equal to zero. These zeros represent the \(x\)-intercept of the function that sets the polynomial equal to zero and can be used to make a rough graph of the function.

Identities

Polynomial identities are equations stating that one polynomial is equal to another. Factoring patterns such as \((a + b)^{2} = a^{2} + 2ab + b^{2}\) are examples of polynomial identities. These identities help you solve problems by converting an expression from a form that looks unsolvable (such as a quadratic expression) into a form that is easier to understand (such as factors).

The Remainder Theorem

The polynomial remainder theorem is useful in certain problems involving polynomial division. If a polynomial \(p(x)\) is divided by \((x-a)\), where a is a number, then the remainder of that division is equal to \(p(a)\). So, if you simply need to know the remainder for a polynomial division by \((x-a)\), you don’t have to actually do the division, just evaluate the polynomial with the value \(x=a\).

For example, if you divided \(x^{3} + 2x^{2} - 5x +3\) by \((x-2)\), the remainder would be 9. If you simply plug \(2\) into the polynomial, you get \(9\) and you don’t have to do the division. The polynomial remainder theorem says that this is true for any polynomial.

Other Math Abilities Tested

While we provide 5 separate math study guides for the SBAC test that cover major areas of math instruction, this test also requires you to do other things in math. We have summarized these at the end of the last page of our SBAC High School Mathematics: Numbers and Operations Study Guide that can be found here. You should go over these concepts, as well.

Tackling Differently Formatted Test Items

There is important information about differently formatted test items on the SBAC exam. Go to our home page for the SBAC to read it as you prepare. Scroll down to “Tips and Tricks.”