Changes

Before continuing, let's see how we actually solved that.  We subtracted 4 from each side of the equation.  On the left side, subtracting 4 canceled the "+4", so we got "x+4-4", which is, of course, just x.  On the right side, we got 7-4, which is 3.  We used the principle that

Before continuing, let's see how we actually solved that.  We subtracted 4 from each side of the equation.  On the left side, subtracting 4 canceled the "+4", so we got "x+4-4", which is, of course, just x.  On the right side, we got 7-4, which is 3.  We used the principle that

::You can apply mathematical operations, to something, even if it contains unknowns like "x".

::You can apply mathematical operations to something, even if it contains unknowns like "x".

The goal was to get an equation that has just "x" on the left side, with some number on the right.  In the original problem, x was on the left side, but hidden with some other mathematical operations.  Our job was to strip off those operations, one at a time.  Here's a more complicated example:

The goal was to get an equation that has just "x" on the left side, with some number on the right.  In the original problem, x was on the left side, but hidden with some other mathematical operations.  Our job was to strip off those operations, one at a time.  Here's a more complicated example:

::<math>\left(\frac{\sqrt{x+31}}{3} + 1\right) \times 6 = 18\,</math>

::<math>\left(\frac{\sqrt{x+31}}{3} + 1\right) \times 6 = 18\,</math>

::<math>3 \times x + 4 \times x = 56\,</math>

::<math>3 \times x + 4 \times x = 56\,</math>

At this point we use the ''distributive law'', along with the principle that

At this point we use the ''distributive law'', along with the principle that

::The various laws of mathematics (commutative, associative, distributive) work even when the items appearing in the expressions are unknowns.

::The various laws of mathematics (commutative, associative, distributive) even when the items appearing in the expressions are unknowns.

The distributive law tells us that

The distributive law tells us that

::<math>3 \times x + 4 \times x = (3+4) \times x\,</math>

::<math>3 \times x + 4 \times x = (3+4) \times x\,</math>