We want to simplify the expressions
by clearing the parentheses.
To clear the parentheses,
we will use the distributive property,
which is often referred to as multiplication
across addition or subtraction.
For the first expression, we have four times
the quantity x minus three.
To clear the parentheses, we distribute four,
which means you multiply four by both terms
inside the parentheses.
So distributing, we have four times x
and then minus four times three.
Four times x is four x, four times three is 12.
The expression simplifies to four x minus 12.
Now I do want to mention, this is formally what's happening,
but often we think of the two products
as four times x, which is four x,
and then because of the subtraction,
we can think of the next product
as four times negative three.
Remember minus three is equivalent to plus negative three.
And since four times negative three is negative 12,
instead of writing plus negative 12,
we write minus 12, which is equivalent.
For the next product, we have negative five
times the quantity x plus seven.
To clear the parentheses, we distribute negative five.
Formally we have negative five times x
plus negative five times seven.
Negative five times x is negative five x.
And then we have plus negative five times seven
is negative 35.
But adding negative 35 is equivalent
to minus positive 35,
and therefore the expression simplifies
to negative five x minus 35.
So formally this is what's happening
when we distribute negative five, but we often
just think of the two products as negative five times x
is negative five x, and the next product
is negative five times positive seven,
which is negative 35.
Adding negative 35 is equivalent to just minus 35.
So if the product is negative, we have subtraction.
If the product is positive, we would have addition.
Next we have negative three times the quantity
two x minus four y.
We distribute negative three.
Formally, this gives us negative three times two x
minus negative three times four y.
Multiplying, negative three times two x is negative six x,
and then we have minus negative three times four y
is negative 12y.
Subtracting a negative is equivalent to adding a positive.
Minus negative 12y is equivalent to plus 12y.
The expression simplifies to negative six x plus 12y.
Again we often don't show this much work
and just think of the two products as negative three
times two x is negative six x,
and then for the next product,
because of the minus four y,
we think of the product as negative three
times negative four y, which is positive 12y,
giving us plus 12y.
So again, when we have subtraction,
we can think of the term as being negative.
And if the product is positive, we have addition.
If the product is negative, we have subtraction.
The last expression, we can think of this two ways.
We can read this as the opposite of the quantity
five x minus three.
To find the opposite of an expression,
we simply change the sign of the terms.
The opposite of five x minus three
is negative five x plus three.
Notice how the sign of both terms has changed.
However we can also clear the parentheses
by distributing a negative one.
So again the original expression was the opposite
of the quantity five x minus three.
We can place a one here and then distribute negative one.
Distributing negative one gives us negative one
times five x minus negative one times three.
Simplifying we have negative five x minus negative three.
Subtracting negative three is equivalent
to adding positive three,
giving us negative five x plus three.
But again in practice, we can just think
of the two products as negative one times five x,
which is negative five x, and for the next product,
because of the subtraction here, we can think
of the product as negative one times negative three,
which is positive three, giving us plus three.
Thinking of it this way allows us to skip this work here.
I hope you found this helpful.