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.

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 ✓.

This problem presents equations 2x - 3y = 6 and 4x - 6y = 12 and asks how many solutions exist. Notice that if we multiply the first equation by 2, we get 4x - 6y = 12, which is exactly the second equation. This means both equations represent the same line, so every point on this line is a solution - there are infinitely many solutions. The student's error was assuming different coefficients mean different lines, but proportional equations (where one is a multiple of the other) represent the same line. When two equations in a system are equivalent, the system has infinitely many solutions.

The question asks for the intersection of y=3 (horizontal) and y=-x+5. Set 3 = -x +5, x=5-3=2, so (2,3). Choices (1,3)(2,3)(3,2)(3,1). Marked B (2,3) yes? Wait, (2,3): y=3, and -2+5=3 yes. C is (3,2): -3+5=2 yes, but horizontal is y=3, 2≠3 no. Marked B yes.

The question asks for the number of $5 bills (f) and $10 bills (t) given 38 total bills worth $275. We set up the system f + t = 38 and 5f + 10t = 275, using substitution by solving the first for f = 38 - t. Substituting into the second gives 5(38 - t) + 10t = 275, simplifying to 190 + 5t = 275, so 5t = 85 and t = 17; then f = 21. The pair (21, 17) satisfies both equations, as 21 + 17 = 38 and 521 + 1017 = 275. A key error is mixing up the variables or forgetting to divide by 5 in solving for t. As a test-taking strategy, check the total value after finding the numbers to ensure it matches $275.

The question seeks the intersection point (x, y) of the two lines given by their equations. I will use substitution by setting the right sides equal: x + 1 = -2x + 7. Adding 2x to both sides gives 3x + 1 = 7, then 3x = 6, x = 2, and y = 2 + 1 = 3, so (2, 3). Errors can arise from sign mistakes when moving terms, like subtracting instead of adding 2x. Another common issue is plugging x back into the wrong equation. A useful strategy is to graph mentally or check by substituting into both originals.

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 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.

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 ✓.