We want to solve the given linear equation.
Normally when solving a linear equation in one variable, we expect one solution, but
that's now always the case.
Let's attempt to solve this equation and see what happens.
Following the guidelines to solve multi-step equations, the first thing we do for this
equation would be to simplify each side by clearing parentheses and combining like terms.
Notice on the left side, we have 2 constant terms.
On the right side, we have 2 x terms.
So on the left side, 9 minus 5 is equal to 4.
Therefore the left side simplifies to 4 plus 4x or 4x plus 4.
Now on the right side, 8x minus 4x is 4x.
The right side simplifies to 4x plus 14.
The next step is to add or subtract to isolate the variable term.
Notice right now we have variable terms on both sides of the equation.
If we are trying to isolate the variable, we need the variable terms on one side.
So to undo the positive 4x on the right side of the equation, let's subtract 4x on both
sides.
Now we simplify both sides.
On the left side, 4x minus 4x is 0.
The left side simplifies to 4.
So we have 4 equals.
On the right side we also have 4x minus 4x.
The right side simplifies to 14.
Notice in the process of isolating the variable, the variable terms simplified out and we are
left with the equation 4 equals 14.
Well we know 4 equals 14 is false.
So because this equation is false, that means there are no solutions to the original linear
equation.
So we can say the equation has no solutions.
Again in the process of solving the equation, if the variable terms simplify out and the
resulting equation is false, this indicates the equation has no solutions.
Sometimes we say, the solution does not exist or DNE for short.
DNE for does not exist.
Let's look at a second example.
We have the same type of equation, so for the first step we will simplify both sides
of the equation.
On the left we have two x terms.
On the right we have two constant terms.
On the left side, 8x minus 6x is 2x.
The left side simplifies to 2x plus 9.
On the right side, 12 minus 3 is equal to 9.
The right side simplifies to 9 plus 2x or if we want 2x plus 9.
Now looking at the equation closely, notice how the expression on the left is exactly
the same as the expression on the right, which means this is going to be true for any value
of x.
Therefore, there is an infinite number of solutions.
But if we don't recognize this, the next step would be to add or subtract to isolate the
variable term.
So to undo the positive 2x on the right side, we would subtract 2x on both sides.
Now we simplify.
On the left, 2x minus 2x is 0.
The left side simplifies to 9.
So we have 9 equals, on the right, 2x minus 2x is also 0.
Notice how the right side also simplifies to 9.
So once again, in the process of isolating the variable, the variable terms simplified
out, but the resulting equation now is true.
9 equal 9 is always true.
So because this equation is true and there are no variables in it, this indicates the
original equation has an infinite number of solutions.
Which means any value of x is going to satisfy the equation or make the equation true.
Sometimes we write infinite solutions for short.
So when solving a linear equation, we normally expect to have 1 solution, but it is possible
to have an infinite number of solutions as well as no solutions.
If an equation has no solutions, it can be classified as a contradiction equation, so
this equation is a contradiction equation.
If an equation has an infinite number of solutions, as we see here, it can be classified as an
identity equation.
If a linear equation in one variable has 1 solution, it can be classified as a conditional
equation.
I hope you found this helpful.