A system of linear equations means two or more linear equations. (In plain speak: 'two or more lines') If these two linear equations intersect, that point of intersection is called the solution to the system of linear equations.
Answer: A system of equation just means 'more than 1 equation.'. A system of linear equations is just more than 1 line, see the picture:
Ok, so what is the solution of a system of equations?
Answer: The solution is where the equations 'meet' or intersect. The red point on the right is the solution of the system.
How many solutions can systems of linear equations have?
There can be zero solutions, 1 solution or infinite solutions--each case is explained in detail below. Note: Although systems of linear equations can have 3 or more equations,we are going to refer to the most common case--a stem with exactly 2 lines.
Case I: 1 Solution
This is
the most common situation and it involves lines that intersect exactly 1 time.
This is the rarest case and only occurs when you have the same line.
Consider, for instance, the two lines below (y = 2x+1 and 2y = 4x +2). These two equations are really the same line .
Example of a system that has infinite solutions:
Line 1: y = 2x + 1
Line 2: 2y = 4x + 2
Example 1
The solution of the system of equations on the left is (2,2) which marks the point where the two lines intersect.
How can we find solutions to systems of equations?
To find the solution to systems of linear equations, you can any of the methods below:
graph : by looking at where lines intersect (meet) on a graph
substitution : by solving for one of the variables and substituting its value in to the other equation.
Elimination : Elimination involves algebraic manipulations of two or more equations. The end goal
is to eliminate a variable by creating opposite coefficients (The examples below should clarify this
straightforward approach).
Use a graphing calculator to find the intersections of the graphs. Yeah, a calculator can do pretty much all of the work for you.
Interactive System of Linear Equations
The Graph Method
Example 1
What is the solution of the following system of equations?
$$
y = x + 1
\\
y = 2x
$$
Use the graph method to solve the system of equations below
$
y = 2x +1
\\
y = 4x -1
$
Step 1
Step 1 is to Graph both equations
Answer
The solution of this system is the point of intersection : (1,3).
Solve the following system of linear equations by graphing.
$
2y = 4x + 2
\\
2y = -x + 7
$
Answer
Step 1) rewrite each equation in slope intercept form.
$
2y = 4x + 2
\\
\frac{1}{2} 2y = \frac{1}{2}(4x+2)
\\
y = 2x +1
\\
2y = 8x - 2
\\
\frac{1}{2} 2y = \frac{1}{2}( 8x - 2)
\\
y = 4x +1
$
This system of lines is the same system that we looked at in the last example.
Step 2 Graph each equation to find the point of intersection --which is the solution. (same as earlier problem)
Let's re-examine system pictured on the left.
$
\color{red}{y} = 2x + 1 \text{ and } \color{red}{y} = 4x -1
$
Step 1) We are going to use substitution like we did in review example 2 above
Now we have 1 equation and 1 unknown, we can solve this problem as the work below shows.
The last step is to again use substitution, in this case we know that x = 1 , but in order to find the y value of the solution, we just substitute x =1 into either equation.
$$
y = 2x + 1
\\
y = 2\cdot \color{red}{1} + 1 = 2 + 1 =3
\\
\text{ or you can substitute x =1 into the other equation}
\\
y = 4x -1
\\
y = 4\cdot \color{red}{1}- 1 = 4 - 1 = 3
\\
\text{solution = }( 1,3)
$$
Substitution Example 2
What is the solution of the system of equations:
$
y = 2x + 1
\\
2y = 3x - 2
$
Step 1) Identify the best equation for substitution and then substitute into other equation.
Step 2) Solve for x
Step 3) Substitute the value of x (-4 in this case) into either equation
$
y = 2x + 1
\\
y = 2\cdot-4 + 1 = -8 + 1 = -7
\\
2y = 3x - 2\\
2y = 3\cdot-4 -2
\\
\text{or you can use the other equation}
\\
2y = -12 -2
\\
2y = -14
\frac{1}{2}\cdot2y =\frac{1}{2}\cdot-14
\\
y = -7
$
The solution of this system is (-4,-7)
Show Graph
Substitution Practice Problems
Solve the system below using substitution
$$
y = x+1
\\
y = 2x +2
$$
Answer
The solution of this system is the point of intersection : (-3,-4).
The Elimination Method
You use elimination when you perform an operation on 1 equation then add the equations so that one of the variables cancels.
Elimination Example 1
$$
y = x + 1
\\
y = -x
$$
Elimination Example 2
Practice Problems
Problem 1)
Use substitution to solve the following system of linear equations:
Line 1: y=3x – 1
Line 2: y= x – 5
Step 1
Set the Two Equations Equal to each other then solve for x
Step 2
Substitute the x value, -2, into either equation to determine y coordinate of solution
$$
y = x -5
\\
y = -2 -5 = -7
$$
The solution is the point (-2, -7) problem 2)
Use the substitution method to solve the system:
Line 1: y = 5x – 1
Line 2: 2y= 3x + 12
Answer
This system of lines has a solution at the point (2, 9).
These lines have the same slope (slope =1) so they never intersect.
Use the substitution method to solve the system:
Line 1: y= x + 1
Line 2: 2y= 3x
Answer
The solution of this system is (1,3).
Use substitution to solve the system:
Line 1: y = 3x + 1
Line 2: 4y = 12x + 3
Answer
Whenever you arrive at a contradiction such as 3 = 4, your system of linear equations has no solutions.
When you use these methods (substitution, graphing etc...) to find the solution what you're really asking is at what
point (which x coordinate and y coordinate) do the two lines intersect. However, if the only time that they intersect
is when 3 = 4, they are never going to intersect since 3 does not equal four!
