Equation Rearranger Notes

Approach

Step 1

Step 1 - Set up the problem. Read the equation and note which variable is to become the subject.

In this problem: The equation is \(y = m \cdot x + c\) and \(c\) is to become the subject. An equation reads the same from either side, so the side holding \(c\) is written first, giving \(m \cdot x + c = y\).

Step 2

Step 2 - Expand and simplify. Remove the brackets and combine the like terms on each side with a Simplifying of Algebraic Expressions.

In this problem: Neither \(m x + c\) nor \(y\) has a bracket to remove or a pair of like terms to combine, so both sides are carried down unchanged.

Step 3

Step 3 - Collect the subject terms. Bring every term that holds the subject onto one side, multiplying both sides up first if the subject sits underneath a division.

In this problem: The right side is \(y\), which holds no \(c\) term, and no side has \(c\) underneath a division, so every subject term already sits on the left and both sides are carried down unchanged.

Step 4

Step 4 - Collect the remaining terms. Move every term without the subject onto the other side.

In this problem: The left side is \(m x + c\), which still holds \(m x\) without \(c\).

Step 4a

Step 4a - Collect the remaining terms. Move every term without the subject onto the other side.

In this problem: \(m x\) is subtracted from both sides to clear \(m x\) off the left.

Step 4b

Step 4b - Collect the remaining terms. Move every term without the subject onto the other side.

In this problem: On the left, \(m x - m x = 0\). On the right, \(y - m x = y - m x\).

Step 5

Step 5 - Factor the subject. When the subject appears in more than one term, write it once outside a bracket so the whole side carries a single subject.

In this problem: \(c\) appears in the single term \(c\), so there is nothing to write outside a bracket and both sides are carried down unchanged.

Step 6

Step 6 - Isolate the subject. Undo whatever still surrounds the subject, so it stands on its own.

In this problem: \(c\) is multiplied by \(1\), so it already stands on its own and both sides are carried down unchanged.

Step 7

Step 7 - Verify the rearrangement. Put the new form back into the original equation and check that both sides still match.

In this problem: Putting \(c = y - m x\) back into the original equation replaces every \(c\). The left side becomes \(y\) and the right side becomes \(m \cdot x + \left(y - m x\right)\), and both simplify to \(y\), so the rearrangement is correct.

Final Answer

Approach

Step 1

Step 1 - Set up the problem. Read the equation and note which variable is to become the subject.

In this problem: The equation is \(v = u + a \cdot t\) and \(t\) is to become the subject. An equation reads the same from either side, so the side holding \(t\) is written first, giving \(u + a \cdot t = v\).

Step 2

Step 2 - Expand and simplify. Remove the brackets and combine the like terms on each side with a Simplifying of Algebraic Expressions.

In this problem: Neither \(u + a t\) nor \(v\) has a bracket to remove or a pair of like terms to combine, so both sides are carried down unchanged.

Step 3

Step 3 - Collect the subject terms. Bring every term that holds the subject onto one side, multiplying both sides up first if the subject sits underneath a division.

In this problem: The right side is \(v\), which holds no \(t\) term, and no side has \(t\) underneath a division, so every subject term already sits on the left and both sides are carried down unchanged.

Step 4

Step 4 - Collect the remaining terms. Move every term without the subject onto the other side.

In this problem: The left side is \(u + a t\), which still holds \(u\) without \(t\).

Step 4a

Step 4a - Collect the remaining terms. Move every term without the subject onto the other side.

In this problem: \(u\) is subtracted from both sides to clear \(u\) off the left.

Step 4b

Step 4b - Collect the remaining terms. Move every term without the subject onto the other side.

In this problem: On the left, \(u - u = 0\). On the right, \(v - u = v - u\).

Step 5

Step 5 - Factor the subject. When the subject appears in more than one term, write it once outside a bracket so the whole side carries a single subject.

In this problem: \(t\) appears in the single term \(a t\), so there is nothing to write outside a bracket and both sides are carried down unchanged.

Step 6

Step 6 - Isolate the subject. Undo whatever still surrounds the subject, so it stands on its own.

In this problem: The left side is \(a t\), so \(t\) is still multiplied by \(a\).

Step 6a

Step 6a - Isolate the subject. Undo whatever still surrounds the subject, so it stands on its own.

In this problem: \(a\) multiplies \(t\), and multiplying is undone by dividing, so both sides are divided by \(a\).

Step 6b

Step 6b - Isolate the subject. Undo whatever still surrounds the subject, so it stands on its own.

In this problem: On the left, \(\frac{a t}{a} = t\), leaving \(t\) on its own. On the right, \(\frac{v - u}{a} = \frac{v - u}{a}\).

Step 7

Step 7 - Verify the rearrangement. Put the new form back into the original equation and check that both sides still match.

In this problem: Putting \(t = \frac{v - u}{a}\) back into the original equation replaces every \(t\). The left side becomes \(v\) and the right side becomes \(u + a \cdot \left(\frac{v - u}{a}\right)\), and both simplify to \(v\), so the rearrangement is correct.

Final Answer

Approach

Step 1

Step 1 - Set up the problem. Read the equation and note which variable is to become the subject.

In this problem: The equation is \(A = l \cdot w\) and \(w\) is to become the subject. An equation reads the same from either side, so the side holding \(w\) is written first, giving \(l \cdot w = A\).

Step 2

Step 2 - Expand and simplify. Remove the brackets and combine the like terms on each side with a Simplifying of Algebraic Expressions.

In this problem: Neither \(l w\) nor \(A\) has a bracket to remove or a pair of like terms to combine, so both sides are carried down unchanged.

Step 3

Step 3 - Collect the subject terms. Bring every term that holds the subject onto one side, multiplying both sides up first if the subject sits underneath a division.

In this problem: The right side is \(A\), which holds no \(w\) term, and no side has \(w\) underneath a division, so every subject term already sits on the left and both sides are carried down unchanged.

Step 4

Step 4 - Collect the remaining terms. Move every term without the subject onto the other side.

In this problem: The left side is \(l w\), where every term holds \(w\), so there is nothing without the subject to move and both sides are carried down unchanged.

Step 5

Step 5 - Factor the subject. When the subject appears in more than one term, write it once outside a bracket so the whole side carries a single subject.

In this problem: \(w\) appears in the single term \(l w\), so there is nothing to write outside a bracket and both sides are carried down unchanged.

Step 6

Step 6 - Isolate the subject. Undo whatever still surrounds the subject, so it stands on its own.

In this problem: The left side is \(l w\), so \(w\) is still multiplied by \(l\).

Step 6a

Step 6a - Isolate the subject. Undo whatever still surrounds the subject, so it stands on its own.

In this problem: \(l\) multiplies \(w\), and multiplying is undone by dividing, so both sides are divided by \(l\).

Step 6b

Step 6b - Isolate the subject. Undo whatever still surrounds the subject, so it stands on its own.

In this problem: On the left, \(\frac{l w}{l} = w\), leaving \(w\) on its own. On the right, \(\frac{A}{l} = \frac{A}{l}\).

Step 7

Step 7 - Verify the rearrangement. Put the new form back into the original equation and check that both sides still match.

In this problem: Putting \(w = \frac{A}{l}\) back into the original equation replaces every \(w\). The left side becomes \(A\) and the right side becomes \(l \cdot \left(\frac{A}{l}\right)\), and both simplify to \(A\), so the rearrangement is correct.

Final Answer

Approach

Step 1

Step 1 - Set up the problem. Read the equation and note which variable is to become the subject.

In this problem: The equation is \(F = m \cdot a + k\) and \(a\) is to become the subject. An equation reads the same from either side, so the side holding \(a\) is written first, giving \(m \cdot a + k = F\).

Step 2

Step 2 - Expand and simplify. Remove the brackets and combine the like terms on each side with a Simplifying of Algebraic Expressions.

In this problem: Neither \(a m + k\) nor \(F\) has a bracket to remove or a pair of like terms to combine, so both sides are carried down unchanged.

Step 3

Step 3 - Collect the subject terms. Bring every term that holds the subject onto one side, multiplying both sides up first if the subject sits underneath a division.

In this problem: The right side is \(F\), which holds no \(a\) term, and no side has \(a\) underneath a division, so every subject term already sits on the left and both sides are carried down unchanged.

Step 4

Step 4 - Collect the remaining terms. Move every term without the subject onto the other side.

In this problem: The left side is \(a m + k\), which still holds \(k\) without \(a\).

Step 4a

Step 4a - Collect the remaining terms. Move every term without the subject onto the other side.

In this problem: \(k\) is subtracted from both sides to clear \(k\) off the left.

Step 4b

Step 4b - Collect the remaining terms. Move every term without the subject onto the other side.

In this problem: On the left, \(k - k = 0\). On the right, \(F - k = F - k\).

Step 5

Step 5 - Factor the subject. When the subject appears in more than one term, write it once outside a bracket so the whole side carries a single subject.

In this problem: \(a\) appears in the single term \(a m\), so there is nothing to write outside a bracket and both sides are carried down unchanged.

Step 6

Step 6 - Isolate the subject. Undo whatever still surrounds the subject, so it stands on its own.

In this problem: The left side is \(a m\), so \(a\) is still multiplied by \(m\).

Step 6a

Step 6a - Isolate the subject. Undo whatever still surrounds the subject, so it stands on its own.

In this problem: \(m\) multiplies \(a\), and multiplying is undone by dividing, so both sides are divided by \(m\).

Step 6b

Step 6b - Isolate the subject. Undo whatever still surrounds the subject, so it stands on its own.

In this problem: On the left, \(\frac{a m}{m} = a\), leaving \(a\) on its own. On the right, \(\frac{F - k}{m} = \frac{F - k}{m}\).

Step 7

Step 7 - Verify the rearrangement. Put the new form back into the original equation and check that both sides still match.

In this problem: Putting \(a = \frac{F - k}{m}\) back into the original equation replaces every \(a\). The left side becomes \(F\) and the right side becomes \(m \cdot \left(\frac{F - k}{m}\right) + k\), and both simplify to \(F\), so the rearrangement is correct.

Final Answer