To solve the given system of equations using the substitution method, we need to isolate one variable in one equation and substitute it into the other equation. Let's go step-by-step:
1. Start with the first equation:
7x - 2y = 77 - 2x
2. Solve this equation for one variable, preferably x or y. Let's solve it for x:
7x + 2x = 77 + 2y
9x = 77 + 2y
3. Next, isolate x by dividing both sides of the equation by 9:
x = (77 + 2y) / 9
4. Now substitute this expression for x into the second equation:
4x + 6y = x + 5y + 29
Replace x with (77 + 2y) / 9:
4((77 + 2y) / 9) + 6y = ((77 + 2y) / 9) + 5y + 29
5. Simplify the equation by getting rid of the fractions. Multiply both sides of the equation by 9 to eliminate the denominator:
4(77 + 2y) + 54y = 77 + 2y + 45y + 261
6. Distribute the multiplication:
308 + 8y + 54y = 77 + 2y + 45y + 261
7. Combine like terms:
62y + 308 = 77 + 47y + 261
8. Simplify further:
62y + 308 = 338 + 47y
9. Move all the y terms to one side of the equation by subtracting 47y from both sides:
62y - 47y + 308 = 338
Simplify:
15y + 308 = 338
10. Finally, solve for y by subtracting 308 from both sides:
15y = 338 - 308
15y = 30
11. Divide both sides of the equation by 15 to isolate y:
y = 30 / 15
y = 2
12. Substitute the value of y back into one of the original equations to solve for x. Let's use the first equation:
7x - 2(2) = 77 - 2x
Simplify:
7x - 4 = 77 - 2x
13. Combine like terms by adding 2x to both sides:
7x + 2x - 4 = 77
Simplify:
9x - 4 = 77
14. Add 4 to both sides of the equation:
9x = 81
15. Divide both sides by 9 to solve for x:
x = 81 / 9
x = 9
Therefore, the solution to the system of equations is x = 9 and y = 2.