Sergeant 6-2 Solving Systems By Substitution Quiz

Explanation
To solve the system by substitution, we can rearrange the second equation to solve for x in terms of y. The equation becomes x = 8 - y. We can then substitute this expression for x into the first equation. The first equation becomes y = 7 - (8 - y)². Simplifying this equation gives us y = 7 - (64 - 16y + y²). Further simplification yields y = 7 - 64 + 16y - y². Combining like terms gives us y² - 15y + 57 = 0. Factoring this quadratic equation gives us (y - 3)(y - 19) = 0. Therefore, the possible values for y are 3 and 19. However, substituting y = 19 back into the second equation gives us a negative value for x, which is not possible. Therefore, the only valid solution is y = 3. Hence, the y-coordinate of the solution is 3.

Explanation
To solve the system of equations by substitution, we can solve one equation for one variable and substitute it into the other equation. From the second equation, we can solve for x in terms of y: x = (7y - 7) / 2. Substituting this expression for x into the first equation, we get (7y - 7) / 2 + 14y = 84. Simplifying this equation, we have 7y - 7 + 28y = 168. Combining like terms, 35y - 7 = 168. Adding 7 to both sides, we get 35y = 175. Dividing both sides by 35, we find y = 5. Therefore, the y-coordinate of the solution is 5.

Explanation
To solve the system of equations by substitution, we can start by solving the second equation for y. From the second equation, we have y = -x - 2. We can substitute this expression for y into the first equation, giving us 3x + (-x - 2) = 12. Simplifying this equation, we get 2x - 2 = 12. Solving for x, we find x = 7. Substituting this value of x back into the second equation, we find y = -9. Therefore, the solution to the system of equations is (7, -9).

Explanation
The given system of equations is y = 2x + 6 and 2x - y = 2. To find the solution, we can substitute the value of y from the first equation into the second equation. Substituting y = 2x + 6 into 2x - y = 2, we get 2x - (2x + 6) = 2, which simplifies to -6 = 2. Since this equation is not true, there is no solution to the system of equations.

Explanation
The given system of equations can be solved using substitution. By rearranging the first equation, we can express x in terms of y as x = 34 + 2y. Substituting this value of x into the second equation, we get 34 + 2y - 8y = 12. Simplifying this equation, we find -6y = -22, which implies y = 22/6 = 11/3. Substituting this value of y back into the first equation, we find x = 34 + 2(11/3) = 34 + 22/3 = 136/3. Therefore, the solution to the system of equations is (136/3, 11/3), which is not one of the given answer choices. Thus, the correct answer is infinitely many solutions.