Answer/Step-by-step explanation:
3. By substitution method, let's substitute [tex] \frac{2}{3}x- 4 [/tex] for y in the first equation.
Thus,
[tex] \frac{1}{3}x + 2(\frac{2}{3}x- 4) = 1 [/tex]
Solve for x
[tex] \frac{x}{3} + \frac{4x}{3} - 4 = 1 [/tex]
Add 4 to both sides
[tex] \frac{x}{3} + \frac{4x}{3} - 4 + 4 = 1 + 4 [/tex]
[tex] \frac{x}{3} + \frac{4x}{3} = 5 [/tex]
[tex] \frac{x + 4x}{3} = 5 [/tex]
[tex] \frac{5x}{3} = 5 [/tex]
Multiply both sides by 3
[tex] \frac{5x}{3}*3 = 5*3 [/tex]
[tex] 5x = 15 [/tex]
Divide both sides by 5
[tex] x = 3 [/tex]
Now, substitute 3 for x in the equation.
[tex] y = \frac{2}{3}x- 4 [/tex]
[tex] y = \frac{2}{3}(3) - 4 [/tex]
[tex] y = 2 - 4 [/tex]
[tex] y = -2 [/tex]
The solution of the equation is x = 3, y = -2
4. Solving by elimination, let's try to eliminate the x-variable by adding both equation together.
[tex] 3x - 2y = 11 [/tex]
[tex]-3x - y = 4[/tex]
[tex] -3y = 15 [/tex] => [tex] (-3x +(-3x) = 0; -2y +(-y) = -3y; 11 + 4 = 15) [/tex]
Divide both sides by -3 to solve for y
[tex] \frac{-3y}{-3} = \frac{15}{-3} [/tex]
[tex] y = -5 [/tex]
Substitute -5 for y in the first equation to find x
[tex] 3x - 2(-5) = 11 [/tex]
[tex] 3x + 10 = 11 [/tex]
Subtract 10 from both sides
[tex] 3x + 10 - 10 = 11 - 10 [/tex]
[tex] 3x = 1[/tex]
Divide both sides by 3
[tex] \frac{3x}{3} = \frac{1}{3} [/tex]
[tex] x = \frac{1}{3} [/tex]
The solution is [tex] x = \frac{1}{3}, y = -5 [/tex]
