Graphical method of finding solutions.

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

Subtracting the equations gives (x=5), then (5+y=9) gives (y=4). In real life, the meeting point is the intersection.

Correct answer: A. \(\left(4,\frac{10}{3}\right)\)

The intersection point is the common solution of both equations. Adding the equations gives \(3x=12\), so \(x=4\). Substituting this into \(x-3y=-6\) gives \(4-3y=-6\), hence \(y=\frac{10}{3}\). Therefore, the intersection is \(\left(4,\frac{10}{3}\right)\). Option B incorrectly reverses the coordinates. Exam tip: After reading an intersection point from a graph, verify it in both equations.

Correct answer: C. (2x+4y=18)

Dividing (2x+4y=18) by (2) gives (x+2y=9). Same left side with different constants gives distinct parallel lines.

Correct answer: A. (5, 2)

The governing principle is that the point where two graphed lines meet must satisfy both equations. Subtract the second equation from the first: (3x + 2y) − (x + 2y) = 19 − 9. This gives 2x = 10, so x = 5. Substitute x = 5 into x + 2y = 9: 5 + 2y = 9, hence 2y = 4 and y = 2. The intersection is therefore (5, 2), making option A correct. Checking the point gives 3(5) + 2(2) = 19 and 5 + 2(2) = 9. Option B reverses the coordinates. Options C and D fail one or both original equations, so they cannot represent the common point on the graph.

Correct answer: A. (0, 14)

To find the \(y\)-intercept, put \(x=0\). Then \(7(0)+2y=28\), so \(y=14\). Therefore, the line cuts the \(y\)-axis at \((0,14)\). Option B is the \(x\)-intercept because it is obtained by putting \(y=0\). Exam tip: For a \(y\)-intercept, always set \(x=0\).

Correct answer: B. \((10,0)\)

To find the x-intercept, put \(y=0\). Then \(2x-5(0)=20\), so \(2x=20\) and \(x=10\). Therefore, the x-intercept is \((10,0)\). Option D has an incorrect x-coordinate, while options A and C lie on the y-axis. Exam tip: Set \(y=0\) to find the x-intercept.

Correct answer: C. Coincident

Dividing the first equation by (5) gives (x+2y=6). Hence both lines are coincident and give infinitely many solutions.

Correct answer: B. Parallel with no solution

Multiplying the first equation by (2) gives (4x+6y=22), different from the second. Therefore the lines are parallel and have no solution.

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

The intersection of two lines is the ordered pair that satisfies both line equations. Since the terms involving y are identical, subtracting one equation from the other removes 5y and makes the calculation simple. Starting with 2x+5y=16 and subtracting x+5y=13 gives x=3.

Now substitute x=3 into the first equation: 3+5y=13. Therefore 5y=10 and y=2. The lines meet at (3,2), so option A is correct. Checking the second equation gives 2(3)+5(2)=6+10=16, confirming the result. A point such as (2,3) may look similar because its coordinates are reversed, but it gives 2+15=17 in the first equation and therefore is not the intersection.

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

At ( (4,2) ), (4(4)+2=18), but (4+2=6). For a common solution, the point must satisfy both equations.

Correct answer: A. \((-2,7)\)

The intersection point must satisfy both equations. Substituting \(x=-2\) into \(3x+y=1\) gives \(3(-2)+y=1\), or \(-6+y=1\), so \(y=7\). Therefore, the intersection point is \((-2,7)\). Option C has the wrong sign for the \(y\)-coordinate. Exam tip: Every point on the vertical line \(x=-2\) has \(x\)-coordinate \(-2\).

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

Every point on the line \(y=4\) has y-coordinate 4. Substituting this into the other equation gives \(2x+3(4)=22\), so \(2x=10\) and \(x=5\). Therefore, the intersection point is \((5,4)\). Option B incorrectly swaps the x- and y-coordinates. Exam tip: for a horizontal line \(y=c\), the y-coordinate is fixed at \(c\).

Correct answer: A. ( (4,8) )

Subtracting the equations gives (2x=8), so (x=4) and (y=8). In real life, the meeting point is the intersection point.

Correct answer: A. Point \(\left(5,3\right)\)

Substituting \(\left(5,3\right)\) makes both equations true. In graphical method this common point is the solution.

Correct answer: B. Point \(\left(4,3\right)\)

At \(\left(4,3\right)\), \(4+3\left(3\right)=13\), so check options carefully. The correct intersection is \(\left(\frac{24}{7},\frac{27}{7}\right)\).

Correct answer: A. Point \(\left(\frac{24}{7},\frac{27}{7}\right)\)

From \(2x-y=3\), \(y=2x-3\), and substituting in the first equation gives \(x=\frac{24}{7}\). Then \(y=\frac{27}{7}\).

Correct answer: A. Point \(\left(4,2\right)\)

At \(\left(4,2\right)\), \(3\left(4\right)+2\left(2\right)=16\) and \(4+2=6\). If both are true, the point is the intersection.

Correct answer: B. Point \(\left(4,3\right)\)

Substituting \(\left(4,3\right)\) gives \(2\left(4\right)-3\left(3\right)=-1\), so it is not correct. The correct solution is \(\left(\frac{22}{5},\frac{13}{5}\right)\).

Correct answer: A. Point \(\left(\frac{22}{5},\frac{13}{5}\right)\)

Using \(y=7-x\) from \(x+y=7\), we get \(2x-3\left(7-x\right)=1\). Thus \(x=\frac{22}{5}\) and \(y=\frac{13}{5}\).

Correct answer: B. Point \((3,6)\)

The intersection point must satisfy both line equations. Subtracting the second equation from the first gives \(3x=9\), so \(x=3\). Substituting this into \(x+y=9\) gives \(y=6\). Therefore, the intersection point is \((3,6)\). For example, option C satisfies \(x+y=9\), but \(4x+y=21\), so it is incorrect. Exam tip: always verify the obtained point in both equations.

Correct answer: B. Point \(\left(3,2\right)\)

At \(\left(3,2\right)\), \(3-2\left(2\right)=-1\), so it is not correct. The correct solution is \(\left(\frac{18}{7},\frac{23}{7}\right)\).

Correct answer: A. Point \(\left(\frac{18}{7},\frac{23}{7}\right)\)

The first equation gives \(x=2y-4\). Substituting in \(3x+y=11\) gives \(y=\frac{23}{7}\) and \(x=\frac{18}{7}\).

Correct answer: A. Point \(\left(3,3\right)\)

At \(\left(3,3\right)\), \(2\left(3\right)+5\left(3\right)=21\) and \(3+3=6\). This is the common point of both lines.

Correct answer: A. (10, 0) और (0, 6)

To find the x-intercept, put y=0: 3x=30, so x=10 and the point is (10, 0). To find the y-intercept, put x=0: 5y=30, so y=6 and the point is (0, 6). Option B reverses the two intercept values. Exam tip: set y=0 for the x-intercept and x=0 for the y-intercept.

Correct answer: B. Point \((0,-8)\)

To find the \(y\)-intercept, set \(x=0\), because every point on the \(y\)-axis has an \(x\)-coordinate of zero. Thus, \(4(0)-3y=24\), giving \(y=-8\). Therefore, the \(y\)-intercept is \((0,-8)\). Option C is the \(x\)-intercept because it is obtained by setting \(y=0\). Exam tip: set \(x=0\) for the \(y\)-intercept and \(y=0\) for the \(x\)-intercept.