Independent System of Equations Examples

Instructor: Mia Primas

Mia has taught math and science and has a Master's Degree in Secondary Teaching.

You may have heard of independent variables, but did you know that systems of equations can be independent? In this lesson, you'll explore some examples of independent systems of equations, including those found in algebraic equations and graphs.

Overview of Independent Systems of Equations

A system of equations consists of two or more equations that are solved simultaneously. To solve the system, you must find solutions for each variable. A system of equations is considered independent if the graphs of the equations create different lines. Independent systems of equations have one solution that can be found graphically or algebraically.

Example of a Graph

In our first example, we will use a graph of the linear equations. The solution to the system can be found by identifying the x and y coordinates of the point of intersection.

Graph of an Independent System of Equations

This system is independent because the graphs of the equations produce two different lines. If the equations produce the same line, they're dependent.

On the graph, the solution is found by locating the point of intersection, which is (2, 3). This point represents the values of x and y that satisfy both equations. We can check to make sure that they satisfy both equations by substituting the values into each equation.

For the first equation, we get 3 = -2 + 5, which is a true equation because -2 + 5 is equal to -3. This tells us that our solution satisfies the first equation. For the second equation, we get 3 = (1/2)(2) + 2, which is also a true equation. Since our solution satisfies both equations, we know that it is correct.

Example of an Algebraic Equation

When equations are graphed, we can quickly determine if a system is independent or not. We can also tell by examining the equations, as long as we know what to look for. In order for the system to be independent, the equations must have different lines when they are graphed. If the equations have different slopes and/or intercepts, they will create different lines. For our next example, we will identify the slope and intercept of each equation in a system of equations.

Algebraic Example of an Independent System of Equations

At first glance, the equations look similar because they have some of the same numbers. But in order to tell if they will produce different lines, we need to identify the slope and intercept. The first equation is in the form y = mx + b, or slope-intercept form. The coefficient of x is the slope (m), and the constant term is the y-intercept (b). This tells us that in the first equation, the slope is negative eight, and the y-intercept is four.

The second equation can be put in slope-intercept form by subtracting 8x and dividing by four on both sides of the equation: y = -2x. Therefore, the slope is negative two, and the y-intercept is zero.

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