What do you call the shortcut in solving a linear equation?

Answer 1

When students have mastered the linear equation solving process (Algebra 2), it is advised that they use the shortcut to solve multiple-steps linear equations. The concept of the shortcut is the following: when a term moves (transposes) from one side of an equation to the other side, its operation changes to inverse operation.

The inverse of an addition operation is a subtraction operation. The inverse of a multiplication operation is a division operation.

The shortcut is actually a very popular world wide equation solving process.

Example 1. Solve: x - 7 = 5.

x = 5 + 7. (-7 moves to the right side and becomes +7).

Example 2. Solve: x - 4 + 9 = 3.

x = 3 + 4 - 9. (Move -4 and +9 to the right side)

Example 3. Solve: 3x/7 = 13.

x = 13(7)/3. (Move 3 and 7 to the right side)

x = 91/3

ADVANTAGES OF THE SHORTCUT:

1. Avoid double writing numbers or/and letters on both sides of the equation that consumes time and that usually are causes for errors.

Example 4. Solve 5x - 4m + 5 = 2x - 3m + 6. (total 6 terms)

Solving without shortcut:

5x - 4m + 5 = 2x - 3m + 6

- 2x = - 2x

3x - 4m + 5 = - 3m + 6

+ 4m - 5 = + 4m - 5

3x = m + 1.

(3x)/3 = (m + l)/3

x = (m + 1)/3.

Solving with shortcut: Move 2x to left side and move -4m + 5 to right side.

5x - 2x = -3m + 6 + 4m - 5. (total 6 terms)

3x = m - 1

x = (m - 1)/3.

2. It simplifies the solving operations. Before proceeding, look for a "smart move" that avoids cross multiplications and distributive multiplications.

Example 5. Solve 5/7 = 11/3(x - 2) ;

Solution. Don't do cross-multiplication and distributive multiplication. Move (x - 2) to left side and move 5 and 7 to the right side. Keep 3 in place.

(x - 2) = 7(11)/3(5) = 77/15 ;

x = 2 + 77/15 = 107/15.

Example 6. Solve 3/(x - 4) = (m + 3)/(m - 1) ;

Solution. Don't do cross multiplication and distribute multiplication.

Move (x - 4) to right side. Keep 3 in place. Move (m+3) and (m - 1) to the left side. 3(m - 1)/(m + 3) = x - 4 ;

x = 4 + 3(m - 1)/(m + 3) ;

Solving this equation (without shortcut) with cross multiplication and distributive multiplications may take too much time and may commit errors.

3. Can easily check at any time the balance of the equation and the numbers of terms /coefficients. No terms /coefficients should be missing during the transpose. There should be no new terms/coefficients added to the equation before grouping like terms.

Example 7. Solve: 3x - 2a - 3b + 5c = 2x + 3 - 4a + 3b - 4x (total 9 terms)

Solution 3x + 4x - 2x = 3 - 4a + 3b + 2a + 3b - 5c (total 9 terms)

5x = 3 - 2a + 6b - 5c ;

4. The shortcut easily helps transform math and science formulas.

Example 8. Formula: 1/R = 1/r1 + 1/r2 + 1/r3. Solve for r1 in terms of other letters.

Solution. 1/r1 = 1/R - 1/r2 - 1/r3 = (r1r2 - Rr3 - Rr2)/R.r2.r3

r1 = R.r2.r3 /(r1r2 - rr2 - rr3)

Example 9. Formula: I = mne/(mR + nr) ; Solve for R in terms of others.

Solution. First move (mR + nr) to left side and move I to right side:

mR + nr = mne/I ;

mR = mne/I - nr = (mne - Inr)/I

R = n(me - Ir)/mI