On this page we hope to clear up problems you might have with systems
of equations and inequalities and their graphs. Systems of equations
are used a lot in other math-related subjects, such as chemistry. Click
any of the links below or scroll down to start understanding systems of equations
3-Variable systems of equations
Systems of inequalities
Quiz on Systems of Equations and Inequalities
Solving systems of equations graphically is one of the easiest
ways to solve systems of simple equations (it's usually not
very practical for complex equations such as hyperbolas or circles). However,
it is usually covered in elementary algebra (Algebra I) courses. We have
followed this custom on this site. Click here
to learn about solving systems of equations.
1. Problem: Solve the following system: x + y = 11 3x - y = 5 Solution: Solve the first equation for y (you could solve for x - it doesn't matter). y = 11 - x Now, substitute 11 - x for y in the second equation. This gives the equation one variable, which earlier algebra work has taught you how to do. 3x - (11 - x) = 5 3x - 11 + x = 5 4x = 16 x = 4 Now, substitute 4 for x in either equation and solve for y. (We use the first equation below.) 4 + y = 11 y = 7 The solution is the ordered pair, (4, 7).
The last method, addition, is probably the most complicated, but is necessary when dealing with more complex systems, such as systems with three or more variables. The idea behind the addition method is to replace an equation with a combination of the equations in the system. To obtain such a combination, you multiply each equation by a constant and add. You choose the constants so that the resulting coefficient of one of the variables will be 0. Example:
2. Problem: Solve the following system: 5x + 3y = 7 3x - 5y = -23 Solution: Multiply the second equation by 5 to make the x-coefficient a multiple of 5. (This works be- cause it does not change the equation (see the multiplication property).) 15x - 25y = -115 Next, multiply the first equation by -3 and add it to the second equation. This gets rid of the x-term. -15x - 9y = -21 15x - 25y = -115 ----------------- - 34y = -136 Now, solve the second equation for y. Then substitute the result into the first equation and solve for x. -34y = -136 y = 4 5x + 3(4) = 7 5x + 12 = 7 5x = -5 x = -1 The solution is the ordered pair, (-1, 4).
Since you would need a three-dimensional coordinate system to solve systems in three variables, solving graphically is not an option. Substitution would work, but is usually unmanageable. Therefore, we will use the addition method, which is basically the same process as it is with systems in two variables.
1. Problem: Solve the following system: x + y + z = 4 x - 2y - z = 1 2x - y - 2z = -1 Solution: Start out by multiplying the first equation by -1 and add it to the second equation to eliminate x from the second equation. -x - y - z = -4 x - 2y - z = 1 ---------------- -3y - 2z = -3 Now eliminate x from the third equation by multiplying the first equation by -2 and add it to the third equation. -2x - 2y - 2z = -8 2x - y - 2z = -1 ------------------ -3y - 4z = -9 Next, eliminate y from the third equation by multiplying the second equation by -1 and adding it to the third equation. 3y + 2z = 3 -3y - 4z = -9 -------------- -2z = -6 Solve the third equation for z. -2z = -6 z = 3 Substitute 3 for z in the second equation and solve for y. -3y - 2z = -3 -3y - 2(3) = -3 -3y - 6 = -3 -3y = 3 y = -1 Lastly, substitute -1 for y and 3 for z in the first equation and solve for x. x + (-1) + 3 = 4 x + 2 = 4 x = 2 The answer is (2, -1, 3).
The easiest way to solve systems of inequalities is to solve them by graphing. Therefore, it is best if you know how to graph inequalities in two variables (5x - 4y < 13, for example). Example:
1. Problem: Graph y < x. Solution: First graph the equation y = x. However, the line must be drawn dashed because the less than sign tells us the line is not included in the solution. Next, test a point that is located above the line and one that is below the line. Any point you pick above the line, such as (0, 2), y is greater than x, so points above the line are not in- cluded in the solution. Points below the line, such as (3, -3) have a y value that is less than the x value, so all points below the line are included in the solution.
1. Problem: Graph the following system: 2x + y >= 2 4x + 3y <= 12 (1/2) <= x <= 2 y >= 0 Solution: See the figure below.
Take the Quiz on systems of equations and inequalities. (Very useful to review or to see if you've really got this topic down.) Do it!