Unit 5: Notes
Section 5.0: What is a System of Linear Equations, and What does it’s “Solution” represent?
A “system” of equations is a set or collection of equations that you deal with all together at once. Linear equations (ones that graph as straight lines) are simpler than non-linear equations, and the simplest linear system is one with two equations and two variables.
Now consider the following two-variable system of linear equations:
y = 3x – 2
y = –x – 6
Since the two equations above are in a system, we deal with them together at the same time. In particular, we can graph them together on the same axis system, like this: |
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A solution for a single equation is any point that lies on the line for that equation. A solution for a system of equations is any point that lies on each line in the system. For example, the red point at right is not a solution to the system, because it is not on either line: |
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The blue point at right is not a solution to the system, because it lies on only one of the lines, not on both of them: |
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The purple point at right is a solution to the system, because it lies on both of the lines: |
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In particular, this purple point marks the intersection of the two lines. Since this point is on both lines, it thus solves both equations, so it solves the entire system of equation. And this relationship is always true: For systems of equations, “solutions” are “intersections”. You can confirm the solution by plugging it into the system of equations, and confirming that the solution works in each equation.
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Determine whether either of the points (–1, –5) and (0, –2) is a solution to the given system of equations.
y = 3x – 2
y = –x – 6
To check the given possible solutions, I just plug the x– and y-coordinates into the equations, and check to see if they work. Copyright © Elizabeth Stapel 2003-2011 All Rights Reserved
checking (–1, –5):
(–5) ?=? 3(–1) – 2
–5 ?=? –3 – 2
–5 = –5 (solution checks)
(–5) ?=? –(–1) – 6
–5 ?=? 1 – 6
–5 = –5 (solution checks)
Since the given point works in each equation, it is a solution to the system. Now I’ll check the other point (which we already know, from looking at the graph, is not a solution):
checking (0, –2):
(–2) ?=? 3(0) – 2
–2 ?=? 0 – 2
–2 = –2 (solution checks)
So the solution works in one of the equations. But to solve the system, it has to work in both equations. Continuing the check:
(–2) ?=? –(0) – 6
–2 ?=? 0 – 6
–2 ?=? –6
But –2 does not equal –6, so this “solution” does not check. Then the answer is:
only the point (–1, –5) is a solution to the system, DONE!!
(Citation: http://www.purplemath.com/modules/systlin1.htm)
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Section 5.1: Solving Systems by Graphing
3 possible scenarios:
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Independent system: |
Inconsistent system: |
Dependent system: |
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This shows that a system of equations may have one solution (a specific x,y-point), no solution at all, or an infinite solution (being all the solutions to the equation). You will never have a system with two or three solutions; it will always be one, none, or infinitely-many. (Citation: (http://www.purplemath.com/modules/systlin2.htm) )
Graphing Systems Example #1:
First, solve each equation for “y =”.
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Graph the lines.
The slope intercept method of graphing was used in this example. The point of intersection of the two lines, (3,0), is the solution to the system of equations. This means that (3,0), when substituted into either equation, will make them both true. See the check. |
| Check: Since the two lines cross at (3,0), the solution is x = 3 and y = 0. Checking these value shows that this answer is correct. Plug these values into the ORIGINAL equations and get a true result. |
| 4x – 6y = 12 4(3) – 6(0) = 12 12 – 0 = 12 12 = 12 (check) |
2x + 2y = 6 2(3) + 2(0) = 6 6 + 0 = 6 6 = 6 (check) |
Graphing Systems Example #2:
Solve this system of equations graphically.
4x – 6y = 12
2x + 2y = 6
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Systems of Equations may also be referred to as “simultaneous equations”.
Let’s look at an example using a graphical method:
First, solve each equation for “y =”.
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Graph the lines. |
The slope intercept method of graphing was used in this example.
The point of intersection of the two lines, (3,0), is the solution to the system of equations.
This means that (3,0), when substituted into either equation, will make them both true. See the check.
Check: Since the two lines cross at (3,0), the solution is x = 3 and y = 0. Checking these value shows that this answer is correct. Plug these values into the ORIGINAL equations and get a true result.
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Section 5.2: Solving Systems using Substitution
Miss. J’s In-Class Notes on How to Solve a System using
the Substitution Method
CLICK ON THE LINK BELOW:
5.2 Miss. J’s Substitution Notes
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Review of 3 possible solutions, and examples of how to solve systems using the substitution method:
Solve this system of equations using substitution. Check.
3y – 2x = 11
y + 2x = 9
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Systems of Equations may also be referred to as “simultaneous equations”.
Let’s look at an example using the substitution method:
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Section 5.3: Solving Systems using Elimination
Review of 3 possible solutions, and examples of how to solve systems using the elimination method:
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Systems of Equations may also be referred to as “simultaneous equations”.
“Simultaneous” means being solved “at the same time”.
Let’s look at three examples using the “addition” or “subtraction” method for systems of equations:
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There is no stopping us now! Let’s try a more involved problem…. |
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(Citation: http://www.regentsprep.org/Regents/math/ALGEBRA/AE3/AlgSysAdd.htm)
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