The question asks for the number of solutions to the given system of linear equations. I will use the elimination method to analyze consistency. Multiply the first equation by 2: 4x - 6y = 24. Compare to the second equation 4x - 6y = 18; the left sides are identical, but right sides differ (24 ≠ 18), indicating inconsistency. Thus, the system has no solutions as the lines are parallel and distinct. A common error is assuming dependence without checking the constants. Another mistake is solving as if consistent and getting contradictory results. For test-taking, check if one equation is a multiple of the other to determine infinite or no solutions.

We need to determine if this system has no solutions, one solution, or infinitely many solutions. Notice that the second equation $4x - 6y = 20$ is related to the first equation $2x - 3y = 9$. If we multiply the first equation by 2, we get $4x - 6y = 18$, which has the same left side as the second equation but a different right side ($18 ≠ 20$). This means the lines are parallel (same slope) but not the same line, so they never intersect. The system has 0 solutions. A key insight for test-taking: when equations have proportional coefficients on the left but different constants on the right, the system is inconsistent.

We need to find how many notebooks and pens were bought. Using elimination, we can subtract the first equation from the second: (2n + p) - (n + p) = 29 - 18, which gives n = 11. Substituting back into n + p = 18: 11 + p = 18, so p = 7. Therefore, (n, p) = (11, 7). To verify: 11 notebooks at $2 each plus 7 pens at $1 each equals 22 + 7 = $29 ✓. A common mistake is confusing which variable represents which item or setting up the equations incorrectly from the word problem.

We need to find the per-mile rate r using two data points. Setting up equations: 17 = f + 6r and 25 = f + 10r. Using elimination, subtract the first from the second: 25 - 17 = (f + 10r) - (f + 6r), which gives 8 = 4r, so r = 2. This means the taxi charges $2 per mile. To find the flat fee: 17 = f + 6(2), so f = 5. A common error is confusing which values go with which distances or making arithmetic mistakes when subtracting equations. Always verify your answer makes sense in the context of the problem.

We need to find the x-coordinate where these two lines intersect. Using elimination, we can add the equations directly since the y terms have opposite signs: $(5x + y) + (2x - y) = 7 + 8$, which gives $7x = 15$, so $x = \frac{15}{7}$. To verify, we can find y: from $5\left(\frac{15}{7}\right) + y = 7$, we get $\frac{75}{7} + y = 7$, so $y = \frac{49}{7} - \frac{75}{7} = -\frac{26}{7}$. A common error is making arithmetic mistakes with fractions. When the answer involves fractions, double-check your arithmetic by substituting back into both original equations.

We need to determine if these equations represent the same line, parallel lines, or intersecting lines. Let me rewrite the first equation in the same form as the second by dividing by 3: x - 2y = 4. This is exactly the same as the second equation x - 2y = 4. Since both equations are identical, they represent the same line, meaning every point on the line is a solution. The common mistake is not simplifying equations to check if they're equivalent. When two equations in a system are identical, the system has infinitely many solutions.

The question asks for the ordered pair (n, d) representing the number of nickels and dimes. I will use the substitution method after setting up the system: n + d = 38 and 0.05n + 0.10d = 2.80 (or 5n + 10d = 280 in cents). Simplify the second to n + 2d = 56. Substitute n = 38 - d into it: 38 - d + 2d = 56, so 38 + d = 56, d = 18, then n = 20. Thus, (n, d) = (20, 18). A key error is forgetting to convert to cents or mishandling decimals. As a strategy, check the total value with the found numbers to verify.

This problem asks how many solutions the system 3x - 2y = 7 and 6x - 4y = 10 has. Notice that if we multiply the first equation by 2, we get 6x - 4y = 14, which has the same left side as the second equation but a different right side (14 ≠ 10). This means the two equations represent parallel lines that never intersect, so the system has no solutions. A common error is thinking that because the second equation's coefficients are multiples of the first, the lines must be the same - but you must check both sides of the equation. When lines are parallel but not identical, the system has 0 solutions.

This problem asks us to find the number of student tickets (x) and adult tickets (y) sold. We'll use the substitution method to solve the system: x + y = 38 (total tickets) and 5x + 8y = 244 (total revenue). From the first equation, x = 38 - y, which we substitute into the second equation: 5(38 - y) + 8y = 244, giving us 190 - 5y + 8y = 244, so 3y = 54, and y = 18. Substituting back, x = 38 - 18 = 20. A common error is mixing up which variable represents which ticket type - remember x is student tickets and y is adult tickets. Always verify your answer by checking both original equations: 20 + 18 = 38 ✓ and 5(20) + 8(18) = 100 + 144 = 244 ✓.

We need to find where y = 2x - 3 and 4x + y = 9 intersect. Since y is already isolated in the first equation, we'll use substitution: substitute y = 2x - 3 into the second equation to get 4x + (2x - 3) = 9. Simplifying: 6x - 3 = 9, so 6x = 12, and x = 2. Substituting back into y = 2x - 3: y = 2(2) - 3 = 1. The warning about sign errors is important - when substituting expressions with subtraction, use parentheses to avoid mistakes. Verify by checking both equations: y = 2(2) - 3 = 1 ✓ and 4(2) + 1 = 9 ✓.