Graphical method of finding solutions.

Correct answer: A. Solution of both equations

The point where both lines meet gives the pair (x,y) satisfying both equations. In exams, always treat the intersection point as the solution.

Correct answer: A. (1) unique solution

One intersection point means the equations have exactly one solution. Remember, intersecting lines are consistent and independent.

Correct answer: A. No solution

Distinct parallel lines never intersect, so there is no common point. In exams, such lines are inconsistent.

Correct answer: A. Infinitely many solutions

Every point on coincident lines satisfies both equations. In exams, call this a consistent and dependent case.

Correct answer: A. ( (2,3) )

For ( (2,3) ), (2+3=5), so it lies on the line. While checking options, substitute (x) and (y) in the equation.

Correct answer: A. ( (3,2) )

In ( (3,2) ), (3-2=1), so it lies on the line. The easy method is to test each option in the equation.

Correct answer: A. 6

Substituting x = 0 in 2x + y = 6 gives 2(0) + y = 6, so y = 6. Thus, the y-intercept of the line is 6 and the corresponding point is (0, 6). Option B is incorrect because when x = 0, the term 2x becomes 0. Exam tip: To find the y-intercept, put x = 0.

Correct answer: A. 8

Substituting \(y=0\) in the equation gives \(x+2(0)=8\), so \(x=8\). Therefore, the x-intercept of the line is the point \((8,0)\). The value 4 may result from confusing the coefficient 2 with the required calculation. Exam tip: To find the x-intercept, always put \(y=0\).

Correct answer: A. ( (3,1) )

The point ( (3,1) ) satisfies both (3+1=4) and (3-1=2). On the graph, this will be the intersection point.

Correct answer: A. (3, 3)

The point of intersection must satisfy both equations. For (3, 3), \(3+3=6\) and \(3-3=0\), so it is the graphical solution. Option B satisfies the first equation but \(6-0\neq0\), so it does not lie on the second line. Exam tip: Substitute the coordinates into both equations to verify the intersection point.

Correct answer: A. x = 2, y = 4

For a pair of linear equations represented by two intersecting lines, the common solution is the coordinate pair of their point of intersection. At the point (2, 4), the horizontal coordinate gives x = 2 and the vertical coordinate gives y = 4. Therefore, the ordered solution is (x, y) = (2, 4), so option A is correct. Option B swaps the coordinates, which changes the point to (4, 2). Options C and D assign the same value to both variables and therefore do not represent the stated intersection. A unique intersection indicates a consistent, independent pair with one solution.

Correct answer: A. ( (2,4) )

The direct answer is option A, the point \((2,4)\). A point \((x,y)\) lies on the line \(y=2x\) exactly when its coordinates satisfy that equation. For option A, substitute \(x=2\): the right side is \(2(2)=4\), which equals \(y=4\). Therefore \((2,4)\) is on the line. Option B, \((3,5)\), fails because \(2(3)=6\), not 5. Option C, \((1,3)\), fails because \(2(1)=2\), not 3. Option D, \((4,2)\), fails because \(2(4)=8\), not 2. The safest method is substitution: take the x-coordinate, multiply it by 2, and compare with the y-coordinate. The line has slope 2, so y must always be twice x. Memory cue: for \(y=2x\), every valid point has the pattern \((x,2x)\).

Correct answer: A. ( (4,5) )

The direct answer is A: \((4,5)\). A point \((x,y)\) lies on \(y=x+1\) when its coordinates satisfy the equation. For A, substitute \(x=4\): the required value is \(y=4+1=5\), exactly the given second coordinate. So A works. For B, \((5,4)\), the equation requires \(y=5+1=6\), not 4. For C, \((2,4)\), the required value is \(2+1=3\), not 4. For D, \((0,2)\), the required value is \(0+1=1\), not 2. The safest method is to substitute each point, or calculate \(y=x+1\) from its x-coordinate. A line contains precisely the points that make its equation true. Do not interchange x and y: the first coordinate is x and the second is y. Memory cue: check whether the second coordinate equals the first coordinate plus one.

Correct answer: A. A vertical line parallel to the (y)-axis

In the equation \(x=3\), the \(x\)-coordinate of every point is 3, while the \(y\)-coordinate can have any value. Hence, all such points lie on a vertical line parallel to the \(y\)-axis. Option B would be correct for \(y=3\). Exam tip: a constant \(x\)-value gives a vertical line, whereas a constant \(y\)-value gives a horizontal line.

Correct answer: A. Horizontal line parallel to the x-axis

In the equation \(y=2\), the y-coordinate is always 2, while x can take any value. Thus, all its points are of the form \((x,2)\), forming a horizontal line parallel to the x-axis. A line parallel to the y-axis has an equation of the form \(x=\text{constant}\). Exam tip: \(y=c\) represents a horizontal line, whereas \(x=c\) represents a vertical line.

Correct answer: A. ( (4,5) )

The line (x=4) has (x=4) for all its points and (y=5) has (y=5) for all its points. Their common point is ( (4,5) ).

Correct answer: A. 5

Substituting \(x=2\) in the equation gives \(2+y=7\). Therefore, \(y=7-2=5\), so option A is correct. The value 7 is the right-hand side of the equation, not the value of \(y\). For the graphical method, the point \((2,5)\) can be plotted on the line.

Correct answer: A. 3

Substituting x = 2 gives 3(2) + y = 9, so 6 + y = 9. Therefore, y = 9 − 6 = 3, and the corresponding point is (2, 3). The value 6 is only the value of 3x, not y. Exam tip: When preparing a table for graphing, substitute the chosen x-value first and then calculate y.

Correct answer: A. (2) points

At least (2) points are enough to draw a straight line. In exams, a third point may be used for checking.

Correct answer: A. (0, 3)

The governing concept is finding an axis intercept from a linear equation. The y-intercept is the point where the graph meets the y-axis, and every point on the y-axis has x-coordinate 0. Substitute x = 0 into x + y = 3: 0 + y = 3, so y = 3. Therefore the y-intercept is (0, 3), making option A correct. Option B has y = 0 and is therefore the x-intercept of this line. The point (1, 1) does not lie on the line because 1 + 1 = 2, not 3. Similarly, (0, 0) gives a sum of 0, not 3. Thus an intercept must satisfy both the equation and the coordinate condition for the relevant axis.

Correct answer: A. (9, 0)

The governing concept is the coordinate condition for an x-intercept. A line meets the x-axis at a point whose y-coordinate is 0. Substitute y = 0 into x + y = 9: x + 0 = 9, so x = 9. Hence the x-intercept is (9, 0), and option A is correct. Option B is the y-intercept because its x-coordinate is 0. The points (1, 8) and (4, 5) do satisfy the equation, since their coordinate sums are 9, but neither lies on the x-axis: their y-coordinates are 8 and 5 rather than 0. This illustrates that merely satisfying the equation is not enough; an x-intercept must also lie on the x-axis.

Correct answer: A. Parallel lines

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

Correct answer: A. Coincident lines

To compare the two lines, simplify the first equation by dividing every term by 2: 2x + 2y = 10 becomes x + y = 5. This is exactly the second equation, not merely an equation with a similar slope. Therefore both equations represent the same geometric line, called coincident lines. Every point on that line satisfies both equations, so the pair has infinitely many solutions. Option A is correct. Distinct parallel lines would have the same slope but different intercepts, while intersecting lines would have different slopes and one common point. Options B, C, and D therefore do not describe this pair.

Correct answer: A. They will intersect at one point

The two equations give different lines with unequal slopes. Hence, they will have one unique intersection point.

Correct answer: A. Inconsistent pair

If two lines have no common point, they are distinct parallel lines, so the pair of equations has no solution. Such a pair is called inconsistent. In contrast, a dependent pair represents coincident lines and has infinitely many solutions. Exam tip: No point of intersection on the graph means an inconsistent pair.