The solution to the equation
is the value of the variable that satisfies the equation
or makes the equation true.
And in order to solve the equation,
we need to isolate the variable on one side of the equation.
And because the given equation has decimals,
we will first simplify both sides of the equation
and then multiply it by a power of 10
to clear the decimals from the equation,
which will give us an equivalent equation without decimals.
So, to begin simplifying on the left,
we will distribute 0.2.
And then on the right, we will clear the parenthesis
by distributing 1.2.
Distributing on the left, 0.2 times x is equal to 0.2 x.
And then we have minus 0.2 times 0.7 is 0.14.
Then we have plus 1.7 x, equals on the right,
distributing 1.2.
1.2 times x is 1.2 x.
And then we have minus 1.2 times 0.3, which is 0.36.
And then we have plus 2.95.
And now let's go ahead and combine like terms.
Notice on the left side we have two x terms.
On the right side, we have two constants.
0.2 x plus 1.7 x is equal to 1.9 x.
The right side simplifies to 1.9 x minus 0.14.
And now combining like terms on the right side.
We have 1.2 x.
And then negative 0.36 plus 2.95
is equal to positive 2.59 giving us plus 2.59.
And at this point, we have a choice
to either keep the decimals in the equation
or clear the decimals from the equation.
In this example, we will clear the decimals
from the equation.
And since we have at most two decimals places,
we clear the decimals from the equation
by multiplying both sides of the equation by 100.
From here, remember the shortcut for multiplying by 100
is to move the decimal point two places to the right.
100 times 1.9 x is 190 x
minus 100 times 0.14 is 14.
On the right, 100 times 1.2 x is 120 x.
And 100 times 2.59 is 259.
And now from here, we need to add or subtract
to isolate the variable terms on one side of the equation.
Notice right now we have x terms on both sides.
Let's begin by undoing the positive 120 x on the right
by subtracting 120 x on both sides.
Notice now the variable terms will be on the left side.
So let's also undo the minus 14 on the left
by adding 14 to both sides.
So here we're combining two steps into one.
Simplifying.
On the left, 190 x minus 120 x is 70 x.
And then negative 14 plus 14 is zero.
This is equal to, on the right,
120 x minus 120 x is zero.
Then we have 259 plus 14,
which is equal to 273.
And at the last step, we multiply or divide to solve for x.
And since 70 x means 70 times x,
to undo the multiplication, we divide both sides by 70.
Simplifying.
70 divided by 70 simplifies to one.
One times x is x.
We have x is equal to 273 divided by 70.
And now we need to find this quotient.
Let's set this up as long division.
We'd begin be determining how many 70s in 273,
which is three.
Three times 70 is 210.
Subtract, the difference is 63.
Because the difference is less than 70.
We know the three is correct.
But now we need to keep going.
So we place the decimal
to the right of the three, the dividend,
move it up into the quotient.
And now can add a zero
to the right of the decimal of the dividend here
and bring the zero down.
And now we need to determine
how many 70s are in 630, which is nine.
Nine times 70 is 630.
We subtract, the difference is zero,
and therefore the quotient is 3.9.
And therefore the solution is x equals 3.9.
And let's verify 3.9 to satisfy the equation
by substituting 3.9 for x
back to the original equation.
Performing this substitution, we have 0.2
times the quantity 3.9 minus 0.7
plus 1.7 times 3.9
is equal to, on the right,
1.2 times the quantity 3.9 minus 0.3
plus 2.95.
And now let's verify the expression on the left
equals the expression on the right using the calculator.
The expression on the left is equal to 7.27.
The expression on the right is also equal 7.27.
7.27 equals a 7.27 is true,
verifying the solution is correct.
I hope you found this helpful.