G.W.Linear Equations

Answer explanation

To solve for x in the equation 3x + 5 = 14, subtract 5 from both sides to get 3x = 9. Then, divide both sides by 3 to find x = 3. Thus, the correct answer is x = 3.

Answer explanation

To solve the system, substitute $x = y + 1$ from the second equation into the first: $2(y + 1) + 3y = 6$. This simplifies to $5y + 2 = 6$, giving $y = 1$ and $x = 2$. The correct solution is (2, 1).

Which of the following systems of equations has no solution?

Answer explanation

The equations \(x + y = 3\) and \(x + y = 5\) represent parallel lines that never intersect, hence they have no solution. The other pairs either represent the same line or intersect at a point.

Answer explanation

To solve the system, substitute y from the second equation into the first: 3x - (3 - 2x) = 7. Simplifying gives 5x = 10, so x = 2. Substituting x back, y = -1. Thus, the solution is (2, -1).

Which of the following systems of equations has infinitely many solutions?

Answer explanation

To solve the system, substitute $x = 5 - 2y$ from the first equation into the second. This gives $3(5 - 2y) - y = 4$. Simplifying leads to $y = 1$ and substituting back gives $x = 2$. Thus, the solution is (2, 1).

Answer explanation

To solve the system, substitute y from the first equation into the second: y = 9 - 4x. This gives 2x - (9 - 4x) = 1, simplifying to 6x = 10, so x = 5/3. Substituting back gives y = 9 - 4(5/3) = 1. Thus, (2, 1) is the solution.

Which of the following systems of equations has exactly one solution?

Answer explanation

To solve the system, substitute y from the first equation into the second. From 5x + 2y = 12, we get y = (12 - 5x)/2. Substituting into 3x - 4y = -6 leads to x = 2, y = 1. Thus, the solution is (2, 1).

Answer explanation

To solve the system, substitute x from the first equation into the second. From x - 3y = -2, we get x = 3y - 2. Substituting into 2x + y = 8 gives 2(3y - 2) + y = 8. Solving yields y = 2, then x = 4. Thus, the solution is (4, 2).