We want to solve the equation
three times the quantity x minus eight
minus six equals five times the quantity x plus six.
The solution to the equation is a value of x
that satisfies the equation, which means
the solution is a value of x that makes the expression
on the left equal to the expression on the right.
In order to solve for x, we need to isolate x
on one side of the equation.
And for this equation, the first step
is to eliminate the parentheses,
which means we'll distribute three here on the left,
and distribute five on the right.
So three times x equals three x,
minus three times eight is 24,
so we have three x minus 24 minus six equals,
distributing five, five times x is five x,
plus five times six is 30, so in the right side we have
five x plus 30.
Next, let's see if we can simplify
either side of the equation.
Notice how we have two like terms here,
negative 24 minus six, or negative 24 plus negative six
equals negative 30, and therefore the left side
simplifies to three x plus negative 30,
or just three x minus 30.
So we have three x minus 30 equals five x plus 30.
In order to isolate x on one side of the equation,
we cannot have an x term on the left,
and an x term on the right, and therefore to isolate x,
or solve for x, we need to either
eliminate or undo the five x on the right,
or eliminate or undo the three x on the left.
Let's undo, or eliminate, the five x on the right,
by subtracting five x on both sides.
Notice, five x minus five x is zero.
And whatever operation we perform to one side
of the equation, we must also perform to the other.
And now let's go ahead and simplify both sides.
Three x minus five x is the same as
three x plus negative five x, which is negative two x.
Negative two x minus 30 equals, on the right,
five x minus five x is zero, that's why we subtracted
five x on both sides, now the right side is just 30.
The next step is to isolate the variable term
by adding or subtracting, which means you want to undo
minus 30, by performing the opposite, or inverse operation.
And therefore, to undo minus 30,
the next step will be to add 30
to both sides of the equation.
Simplifying both sides again, in the left side
adding 30 undoes minus 30, or negative 30 plus 30 is zero,
so now we have the equation negative two x equals,
30 plus 30 is equal to 60.
So now we have the equation negative two x equals 60.
The last step is to multiply or divide
in order to solve for x,
and negative two x means negative two times x,
and therefore to undo multiplying by negative two,
we perform the opposite or inverse operation,
which means we divide both sides by negative two.
Dividing by negative two undoes multiplying by negative two,
and therefore the left side simplifies to just x,
or negative two divided by negative two is one,
and one times x is x.
And x is equal to 60 divided by negative two,
which equals negative 30.
So our solution is x equals negative 30,
but let's check this by making sure
it does satisfy the equation.
To do this, we'll substitute negative 30 for x
in the original equation.
We would have three times the quantity
negative 30 minus eight, minus six
equals five times the quantity negative 30 plus six.
Let's go ahead and check this on the calculator.
So on the left, we have three times the quantity
negative 30, minus eight, minus six, enter.
The left side equals negative 120.
On the right side we have, five times the quantity
negative 30, plus six, but it's also equal to negative 120.
Which means x equals negative 30,
satisfies the equation, and our solution is correct.
I hope you found this helpful.