Graphical method of finding solutions.

Dividing every term of the first equation, \(3x+3y=24\), by 3 gives \(x+y=8\), which is the second equation. Hence, both equations represent the same line and have infinitely many common solutions. Exam tip: If one linear equation is a non-zero constant multiple of the other, their graphs are coincident, or the same line.

Both have the same coefficients of (x) and (y), but different constants. Therefore, they are distinct parallel lines.

The governing concept is generating an ordered pair from a linear equation. Since y = 3, substitute this value into x - y = 8: x - 3 = 8. Adding 3 to both sides gives x = 11. The ordered pair must be written in the order (x, y), so it is (11, 3), which is option B. Option A reverses the coordinates. Option C uses 8 as the x-coordinate without solving the equation, while option D incorrectly subtracts 3 from 8 instead of adding it when isolating x. Substitution verifies the result: 11 - 3 = 8.

For the point (3, 7), substituting x = 3 and y = 7 gives 3x − y = 3(3) − 7 = 9 − 7 = 2. Therefore, the point lies on the line. The other points do not satisfy the equation; for example, (1, 3) gives 3(1) − 3 = 0. In an exam, test a point by directly substituting its coordinates into the equation.

In the point ( (0,12) ), (x=0), so it lies on the (y)-axis. Identifying axes helps in reading graphs.

In the point ( (14,0) ), (y=0), so it lies on the (x)-axis. This is useful in intercept questions.

To find the x-intercept, put \(y=0\), which gives \(x=18\); hence the point is \((18,0)\). To find the y-intercept, put \(x=0\), which gives \(y=18\); hence the point is \((0,18)\). Therefore, option B is correct. The points in option A satisfy \(x+y=9\), not the given equation. Exam tip: set y=0 for the x-intercept and x=0 for the y-intercept.

The governing idea is that two distinct points determine a straight line. For the x-intercept, put y = 0: 3x = 24, so x = 8, giving (8, 0). For the y-intercept, put x = 0: 6y = 24, so y = 4, giving (0, 4). Thus option C gives both correct points; the other pairs contain incorrect intercepts or points not satisfying the equation.

The governing concept is the graphical form of vertical lines. An equation x = a fixes the x-coordinate while allowing every value of y, so it represents a vertical line. Thus x = 5 and x = 12 are two vertical lines with different positions. Vertical lines have the same direction and never meet when their x-values differ, so they are parallel. Option D is correct. They do not intersect because 5 cannot equal 12, and they do not coincide for the same reason. Neither line is the x-axis; the x-axis is y = 0. Therefore the other choices confuse line orientation or position.

The governing concept is the number of common points of two graphs. An equation y = a represents a horizontal line because the y-coordinate is fixed while x may take any value. Therefore y = 3 and y = 11 are distinct horizontal lines. Since their y-values are different, no point can satisfy both equations at the same time: a common point would have to have y equal to 3 and 11 simultaneously. Hence the pair has no solution, so option B is correct. Coincident lines would give infinitely many solutions, while intersecting lines would give one solution; neither situation occurs here.

In an ordered pair (x, y), the first coordinate is the x-coordinate and the second coordinate is the y-coordinate. Therefore, in (7, 13), y = 13. The value 7 is the x-coordinate, not the y-coordinate. Exam tip: the second coordinate of (x, y) always gives the value of y.

In an ordered pair \\( (x,y) \\), the first coordinate represents \\(x\\) and the second represents \\(y\\). Therefore, for the intersection point \\( (15,2) \\), \\(x=15\\). The value 2 in option A is the value of \\(y\\), not \\(x\\). Exam tip: Always read the coordinates in the order \\(x,y\\).

The second equation is (3) times the first, so the lines are coincident. Coincident lines have infinitely many solutions.

Multiplying the first equation by (3) gives (3x-3y=21), which differs from (3x-3y=24). Therefore, the lines are parallel.

Substituting \(x=5\) into the equation \(x+5y=15\) gives \(5+5y=15\). Hence, \(5y=10\) and \(y=2\). Therefore, the point is \((x,y)=(5,2)\). Exam tip: verify a proposed point by substituting both coordinates into the original equation.

Substituting y = 1 in 5x + y = 26 gives 5x + 1 = 26, so 5x = 25 and x = 5. Therefore, the ordered pair is (5, 1). In option B, the x- and y-coordinates are interchanged. As an exam tip, verify a point by substituting both coordinates into the original equation.

Substituting the point \((4,2)\) into both equations gives \(4+2(2)=8\) and \(2(4)+2=10\). Therefore, \((4,2)\) is the common point and hence the graphical intersection of the two lines. The nearby option \((3,2)\) is incorrect because it gives \(3+2(2)=7\), not 8, in the first equation. Exam tip: verify an intersection point in both equations.

The intersection point must satisfy both equations. Subtracting the first equation from the second gives x = 5. Substituting this into the first equation, 5 + 4y = 17, so y = 3. Therefore, the graphical solution is (5, 3). Exam tip: Substitute each option into both equations for a quick check; (3, 5) does not satisfy the first equation.

The governing concept is accurate scale selection in graph construction. A scale tells how numerical coordinate units correspond to physical distances on the graph paper. If the scale is wrong or applied inconsistently, a coordinate such as (2, 5) may be marked at an incorrect horizontal or vertical distance. Consequently, the plotted points do not represent the given values, and the line or intersection may appear in the wrong position. Therefore option A is correct. The algebraic equations do not change merely because the scale is chosen poorly, and naming the line or reading the question is unrelated to the immediate graphical error. A correct scale should be selected before plotting.

The governing convention for an ordered pair is (x, y): the first coordinate gives horizontal displacement and the second gives vertical displacement. Thus (9, 4) means x = 9 and y = 4, whereas (4, 9) means a completely different point with x = 4 and y = 9. Writing one in place of the other reverses the coordinate order, so option B is correct. The error is not about naming axes, because the coordinates themselves have been interchanged. It is not about failing to draw a line, and option D describes a correct action rather than a mistake. Careful use of (x, y) prevents this error.