a) Max = 8

b) Max = 4

SHORTCUT IN SOLVING LINEAR EQUATIONS 

A. GENERALITIES. 

When students have a complete understanding of the linear equation solving process, especially when they get to the Algebra 2 level, it is advised that students use a short cut to solve multiple-step linear equations. By using the shortcut, students can avoid the double writing of coefficients and constants on both sides of the equation. Therefore, it helps students to solve linear equations faster with fewer errors and mistakes.
The shortcut is actually a very popular worldwide process of equation solving. 

B. BASIC CONCEPT. 

The basis concept of the shortcut in solving linear equations is as follows: 

"When a term moves to the other side of the equation, its operation changes to the inverse operation" 

The inverse of an addition operation is a subtraction operation. Multiplication and division are inverse operations. 

C. OPERATION RULE OF THE SHORTCUT. 

When a term moves to the other side of an equation: 

a. Addition (+n) changes to subtraction (-n). Example: 
x + n = 8.
x = 8 - n. (n moves to the right side and becomes -n) 

b. Subtraction (-n) changes to addition (+n). Example:
x - n = 5.
x = 5 + n. (-n moves to the right side and becomes n). 

c. Multiplication (.n) changes to division (1/n). Example:
n.x = 7.
x = 7/n. (n moves to the right side and becomes 1/n) 

d. Division (1/n) changes to multiplication (.n). Example:
x/n = 9.
x = 9n. (1/n moves to the right side and becomes .n) 

D. EXAMPLES OF SOLVING LINEAR EQUATIONS USING THE SHORTCUT. 

First, move all terms containing x to one side and all numbers (or/and letters) to the other side of the equation. Then, after combining terms, move the coefficient of x to the other side to get x alone. 

Example 1. Solve: 3x - 4 = 2x + 5.
Solution. Move 2x to the left side and move -4 to the right side:
3x - 2x = 5 + 4.
x = 9 ; (combining terms). 

Example 2. Solve: 7x - 3 + 5 = 10 + 3x.
Solution. 7x - 3x = 10 + 3 - 5 ; (Moving terms)
4x = 8 ; (Combine terms)
x = 8/4 = 2 ; (Move 4 to the right side). 

Example 3. Solve: 2x - a = b + c - 3x.
Solution. 2x + 3x = a + b + c ; (Moving terms)
5x = a + b + c ; (Combine terms).
x = (a + b + c)/5 ; (Move 5 to the right side). 

Example 4. Solve: mx + 3 = x + m.
Solution. Move x terms to the left side and move all numbers and letters to the right side.
mx - x = m - 3.
x(m - 1) = m - 3. ; (Factoring).
x = (m - 3)/(m - 1); Move the term (m - 1) to the right side. 

NOTE 1. In the above examples, terms containing x are kept on the left side, and simplification is done on the right side. For solving conveniences, students may proceed solving by keeping x-terms on the right side of the equation. The reasons are to avoid moving the x terms and/or to keep the coefficients of x-terms positive. 

Example 5. Solve: 2(x + 1) = 5(x - 2).
Solution. 2x + 2 = 5x - 10 ; (Distributive multiplication).
2 + 10 = 5x - 2x ; (Move terms, and keep coefficient positive)
12 = 3x ; (Combine terms).
4 = x ; (Simplify). 

Example 6. Solve: 9 - x = 3x + 1.
Solution. 9 - 1 = 3x + x ; (Move terms and keep coefficient positive).
8 = 4x ; (Combine terms).
2 = x ; (Simplify). 

Example 7. Solve: 3(m - 1) = 2(x - m).
Solution: 3m - 3 = 2x - 2m ; (Distributive multiplication)
3m + 2m - 3 + 4 = 2x ; ( Move terms and avoid moving x-term).
5m + 1 = 2x ; (Combine terms)
(5m + 1)/2 = x ; (Move coefficient of x to the left side). 

NOTE 2 - "SMART MOVE". 

Operations such as cross multiplication and distributive multiplication are sometimes not necessary.
Before proceeding, students are advised to look for a "Smart Move" in order to save time and to avoid repetition. 

Example 8. Solve: 4/3 = (x - 6)/7.
Solution. Don't automaally do cross multiplication and distributive multiplication to avoid repetition and to save time. Leave (x - 6) in place, and move number 7 to the other side.
x - 6 = (4.7)/3 = 28/3 ; (Move 7 to the other side)
x = 6 + 28/3 = 46/3 ; (Move 6 to right side then combine terms). 

Example 9. Solve: 3/7 = 6/(x - 2).
Solution. Don't proceed cross multiplication and distributive multiplication. Move (x - 2) to the left side and move 3 and 7 to the right side.
x - 2 = (7.6)/3 = 42/3 ; (Move terms and simplify)
x = 2 + 42/3 = 48/3 = 16 ; (Move term and simplify). 

Example 10. Solve: a/b = c/d(x - e).
Solution. Do not proceed cross multiplication and distributive multiplication. Move (x - e) to the left site and move a and b to the right side. Keep d in place.
x - e = bc/ad ; (Move terms).
x = e + bc/ad = (ead + bc)/ad ; (Move terms and common denominator). 

Example 11. Formula: 1/f = 1/d1 + 1/d2. Find d1 in terms of f and d2.
Solution. Solving for d1 proceeds solving a linear equation using shortcut:
1/d1 = 1/f - 1/d2 = (d2 - f)/fd2 ; ---> d1 = fd2/(d2 - f). 

Example 12. Formula 1/R = 1/r1 + 1/r2 + 1/r3. Find r1 in terms of R,r2,and r3.
Solution. Solve for r1 using shortcut in solving linear equation.
1/r1 = 1/R - 1/r2 - 1/r3 = (r2r3 - Rr3 - Rr2)/Rr2r3.
r1 = Rr2r3/(r2r3 - Rr3 - Rr2). 

E. ADVANTAGES OF THE SHORTCUT IN SOLVING MULTIPLE STEPS LINEAR EQUATIONS. 

1. Can avoid the double writing of terms (to be eliminated) on both sides of the equation, in every step of solving. 
Example 13. Solve: x - m + 3 = m(x + 2) - 5 ;
Solution (without shortcut) 
x - m + 3 = mx + 2m - 5 ;
- mx = - mx ;
x - mx - m + 3 = 2m - 5 ; 
+ m = + m ; 
x(1 - m) + 3 = 3m - 5 ;
-3 = -3 ;
x(1 - m) = 3m - 8 ;
: (1 - m) = : (1 - m) ;
x = (3m - 8)/(1 - m) ; 

Example 13 (with short cut):
x - m + 3 = mx + 2m - 5 ;
x - mx = 2m - 5 + m - 3 ;
x(1 - m) = 3m - 8 ;
x = (3m - 8)/(1 - m) ; 

Example 14. Solve: (x - m + 2)/3 = 2(x + 4)/9 ;
Solution (without shortcut):
9(x - m + 2) = 6(x + 4) ;
9x - 9m + 18 = 6x + 24 ;
- 6x = - 6x ;
3x - 9m + 18 = 24 ;
+ 9m = + 9m ;
3x + 18 = 9m + 24 ;
- 18 = - 18 ;
3x = 9m + 6
: 3 = : 3 ;
x = 3m + 2 ; 

Example 14:
Solution (with shortcut)
9(x - m + 2) = 6(x + 4) ;
9x - 9m + 18 = 6x + 24 ;
9x - 6x = 9m + 24 - 18 ;
3x = 9m + 6 ;
x = 3m + 2 ; 

2. Can use the "smart move" to avoid cross multiplication and/or distributive multiplication that are sometimes not necessary. 

Example 15. Solve: m/3 = 2(m - 1)/5(x - 2) ;
Solution (without shortcut):
5m(x - 2) = 6(m - 1) ;
5mx - 10m = 6m - 6 ;
+ 10m = + 10m ;
5mx = 16m - 6 ;
: 5m = : 5m
x = (16m - 6)/5m (Answer) 

Example 15. Solve: m/3 = 2(m - 1)/5(x - 2) ;
Solution (with shortcut): Move (x - 2) to the left side. Move m and 3 to the right side:
x - 2 = 6(m - 1)/5m ;
x = 2 + 6(m - 1)/5m (Answer) 

3. Help easily transform sciences and math formulas. 

Example 16. Transform the formula V2 = V1.R2/(R1 + R2) to get R2 in terms of other letters.
Solution (without shortcut):
V2.(R1 + R2) = V1.R2 (cross multiplication)
V2.R1 + V2.R2 = V1.R2 (Distributive multiplication)
- V1.R2 = - V1.R2 ;
V2.R1 + V2.R2 - V1.R2 = 0 ;
V2.R1 + R2(V2 - V1) = 0 ;
- V2.R1 = - V2.R1 ;
R2.(V2 - V1) = - V2.R1 ;
: (V2 - V1) = : (V2 - V1)
R2 = - V2.R1/(V2 - V1) ; 

Example 16. Transform the formula V2 = V1.R2/(R1 + R2) to get R2 in terms of other letters.
Solution (With shortcut):
V2.(R1 + R2)= V1.R2 
V2.R1 + V2.R2 = V1.R2 (3 terms)
V2.R2 - V1.R2 = -V2.R1 (3 terms)
R2.(V2 - V1) = -V2.R1
R2 = -V2.R1/(V2 - V1) 

Example 17. Transform the formula 1/f = 1/d1 + 1/d2 to get d2 in terms of other letters.
Solution (Without shortcut): 
1/f = 1/d1 + 1/d2 = (d2 + d1)/(d1.d2) (common denominator)
f.(d2 + d1) = d1.d2 (cross multiplication) ;
f.d2 + f.d1 = d1.d2 (Distributive multiplication) ;
f.d2 - d1.d2 + f.d1 = 0 
d2.(f - d1) = -f.d1 ;
d2 = -f.d1/(f - d1); 

NOTE; Errors could be easily committed during double writing terms on both sides of the equations.
Double writing also makes the solving process slower. 

Example 17.
Solution (with shortcut):
1/d2 = 1/f - 1/d1 (Move 1/d1 to left side, then switch side)
1/d2 = (d1 - f)/(f.d1) (common denominator)
d2 = f.d1/(d1 - f) (inverse the equation); Answer. 

F. EXERCISES ON SOLVING LINEAR EQUATIONS USING SHORTCUT. 

1. Coefficients and constants are whole numbers. 

1). 4x - 3 = -2x + 3 ; 2). -2x + 3 = 5 - 3x ; 3). 4(x + 2) = 2x - 10 ; 4). 2(x - 2) = 2(3 - x); 
5). 2(x - 1) + 3(x - 8) = 0 ; 6). 3x - 5 = 4(x - 7) ; 7). (2x - 3)/4 = (7 - x)/5; 
8). 3(x - 1) + 2(x + 4) = 4(x - 6); 9). 2(2x - 3) - 3(x - 5) = 0 ; 10). (3x + 5)/4 = 3 - x. 

2. Coefficients and constants are letters and/or numbers. 

11). 2(x - a) + b = -a + 2b + c ; 12). ax = -bx - cx + d ; 13). (1 - a)x = a(1 + a) ; 
14). x + t = 1 - tx ; 15). a(x - 1) = 2(x + 2) ; 16). tx - t = x + 1 ; 16. ax + ab = -bx + ac ; 
17). 3a + bx = 2(x + a) ; 18). mx - 3 = 2m(x + 5) ; 19). xt^2 - 3 = x - 3t. 
20). (2x - 3)/3 + 1/4 = 5 ; 21). 4/5 + 2x/3 = 3/2 ; 22). 3x/5 = 6 - x/3 ; 23). 2(x - 1/3) = x/4 + 5. 
26). x/3a - 1/b = x/c - 2/d ; 27). ax/2 - c = cx/d - b. 28). mx/3 - 2/5 = x/3 + 3/5. 

3. Review exercises. 
29). 3(x - m) = 5x - m - 2 ; 30). mx = x - 1 + m ; 31). 2ax = c(x - 5) + b ; 
32). ax + b(x - 1) + cx = a ; 33). 2ax = b + c(x - 5) ; 34). t^2.x - t = 3(tx - 5) ; 
35). a^2.x = b^2.x + a - b ; 36). 2x - t = 2(t^2.x + 1) - 1 ; 37). 2x/3 - 9/5 = x/2 ; 
38). 3/x - 4/2x = 11/2 ; 39). 1/x + 3/2 = 2/3x ; 40. ax/b = 3 + cx ; 

\(A=\left(2x+\frac{1}{3}\right)^4-1\) . Có: \(\left(2x+\frac{1}{3}\right)\ge0\)

\(\Rightarrow\left(2x+\frac{1}{3}\right)^4-1\ge-1\)

Dấu = xảy ra khi: \(2x+\frac{1}{3}=0\)

\(\Rightarrow2x=-\frac{1}{3}\)

\(\Rightarrow x=-\frac{1}{3}:2=-\frac{1}{6}\)

Vậy: \(Min_A=-1\) tại \(x=-\frac{1}{6}\)