Graphical method of finding solutions.

Correct answer: A. Point \(\left(\frac{25}{7},\frac{23}{7}\right)\)

Solving both equations gives \(x=\frac{25}{7}\) and \(y=\frac{23}{7}\). On the graph this is the intersection point.

Correct answer: A. Point \(\left(\frac{66}{17},\frac{65}{17}\right)\)

From the second equation, \(x=23-5y\). Substituting this into the first equation gives \(3(23-5y)-2y=4\), or \(69-17y=4\), so \(y=\frac{65}{17}\). Therefore, \(x=23-5\left(\frac{65}{17}\right)=\frac{66}{17}\). Hence, the two lines intersect at \(\left(\frac{66}{17},\frac{65}{17}\right)\). In option B, the value of \(x\) is incorrect. Exam tip: The graphical solution is the common point of the two lines; verify the point by substituting it into both equations.

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

\(\left(2,3\right)\) satisfies both equations. In difficult options, direct substitution is the fastest check.

Correct answer: C. Point \(\left(\frac{106}{19},\frac{69}{19}\right)\)

Elimination gives \(19y=69\) and \(x=\frac{106}{19}\). Fraction coordinates can also be graphical solutions.

Correct answer: B. Parallel and distinct lines

(\frac{a_1}{a_2}=\frac{b_1}{b_2}=2), but (\frac{c_1}{c_2}=\frac{10}{7}). Hence the lines are parallel and inconsistent.

Correct answer: C. Coincident

The direct answer is C: the graphs are coincident, meaning they are the same line. Compare the equations \(6x-9y=15\) and \(2x-3y=5\). Divide every term of the first equation by 3: \((6x-9y)/3=15/3\), giving \(2x-3y=5\), exactly the second equation. Therefore every point satisfying one equation satisfies the other, so there are infinitely many common solutions. Option C is correct. Option A, parallel and distinct, would apply if the coefficients of x and y were proportional but the constants were not proportional. Option B, intersecting at one point, would mean the two equations represented different lines with different slopes. Option D, intersecting at the origin, is not possible here because the origin gives 0=15 in the first equation, which is false. Exam cue: if one complete linear equation is a non-zero multiple of the other, the lines are coincident.

Correct answer: B. (3x-y=5) and (6x-2y=11)

Multiplying the first equation by (2) gives (6x-2y=10), while the second is (6x-2y=11). Hence the lines are parallel and have no solution.

Correct answer: A. \((4,0)\) और \((0,-7)\)

For the x-intercept, put \(y=0\): \(7x=28\), giving \(x=4\), so the point is \((4,0)\). For the y-intercept, put \(x=0\): \(-4y=28\), giving \(y=-7\), so the point is \((0,-7)\). Therefore, option A is correct. Exam tip: check the sign of the y-intercept and maintain the stated order—x-intercept first, y-intercept second.

Correct answer: A. Point \(\left(-4,-11\right)\)

Putting \(x=-4\) gives \(3\left(-4\right)-2y=10\), so \(y=-11\). In a vertical line, (x) is already fixed.

Correct answer: A. \((8,5)\)

The point of intersection must satisfy both line equations. From the first line, \(y=5\). Substituting this into the second equation gives \(4x-3(5)=17\), so \(4x=32\) and \(x=8\). Hence, the intersection point is \((8,5)\). Remember that \(y=5\) is a horizontal line, so its points all have y-coordinate 5.

Correct answer: C. Lines are coincident

When all three ratios are equal, both equations represent the same line. Therefore there are infinitely many common points.

Correct answer: C. There is exactly (1) solution

The direct answer is C: there is exactly one solution. For two lines \(a_1x+b_1y+c_1=0\) and \(a_2x+b_2y+c_2=0\), unequal ratios \(\frac{a_1}{a_2}\ne\frac{b_1}{b_2}\) mean their direction slopes are different. Lines with different slopes cannot remain parallel or identical; they meet at one and only one point. Therefore the pair is consistent and independent and has exactly one solution. Option A, no solution, applies to distinct parallel lines, usually when \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}\). Option B, infinitely many solutions, applies to coincident lines, when all three ratios are equal. Option C is correct. Option D is wrong because unequal coefficient ratios do not make lines identical; identical lines require matching ratios. Exam cue: compare the first two ratios. Unequal means one intersection; equal first two ratios require checking the constant ratio.

Correct answer: B. The lines will be distinct and parallel

When \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}\), the two lines have the same slope but are not the same line. Hence, they are distinct parallel lines and have no common point. Coincident lines require all three ratios to be equal, while intersecting lines require the first two ratios to be unequal. Exam tip: Compare the ratios in the order \(a\), \(b\), and \(c\) to identify the graphical condition.

Correct answer: A. Point \((4,\frac{7}{2})\)

The intersection point is the common solution of both linear equations. Adding the equations gives \(4x=16\), so \(x=4\). Substituting this into \(x+2y=11\) gives \(4+2y=11\), hence \(y=\frac{7}{2}\). Therefore, the intersection point is \(\left(4,\frac{7}{2}\right)\). In option B, the values of \(x\) and \(y\) are interchanged. Exam tip: Always substitute the ordered pair into both equations to verify the intersection point.

Correct answer: B. Point \(\left(3,\frac{5}{2}\right)\)

\(\left(3,\frac{5}{2}\right)\) satisfies both equations. Read fraction coordinates carefully using scale on the graph.

Correct answer: A. Point \(\left(1,5\right)\)

At \(\left(1,5\right)\), \(2\left(1\right)+5\left(5\right)=27\), but \(1+5=6\). To be a common solution, both equations must be true.

Correct answer: B. Treating coincident lines as having one solution

The direct answer is B: the mistake is treating coincident lines as having one solution. Multiplying 2x+3y=12 by 2 gives 4x+6y=24, which is exactly the second equation. Thus both equations represent the same line. Their common points are not limited to one point; every point on that line is a solution, so there are infinitely many solutions. Option A describes distinct parallel lines, which have no common point, so it is wrong. Option B is correct. Option C describes a misunderstanding of intersecting lines; intersecting lines meet at one point and therefore give one consistent solution. Option D concerns coordinate order and has no connection with coincident lines. Memory cue: one equation being a non-zero multiple of the other means the same line and infinitely many solutions.

Correct answer: B. \(x=-\frac{3}{2},\ y=4\)

The first coordinate of a point is (x) and the second is (y). Do not change order while reading negative fraction coordinates.

Correct answer: B. They are parallel and distinct

Multiplying the first equation by (2) gives (6x-4y=12), while the second is (6x-4y=18). Hence the lines are parallel and distinct.

Correct answer: B. Point (9, 0)

The line \(y=0\) represents the \(x\)-axis, so the point of intersection must have \(y=0\). Substituting this into the first equation gives \(2x+3(0)=18\), hence \(2x=18\) and \(x=9\). Therefore, the intersection point is \((9,0)\). Exam tip: To find where a line meets the \(x\)-axis, put \(y=0\).

Correct answer: B. Point \\(0,-4\\)

On the line \\(x=0\\), the x-coordinate of the intersection point must be 0. Substituting \\(x=0\\) in \\(4x-5y=20\\) gives \\(4(0)-5y=20\\), so \\(-5y=20\\) and \\(y=-4\\). Hence, the intersection point is \\((0,-4)\\). Option A has the wrong sign for \\(y\\). Exam tip: the line \\(x=0\\) is the y-axis, so substitute \\(x=0\\) directly into the other equation.

Correct answer: C. Lines are parallel

(\frac{a_1}{a_2}=\frac{b_1}{b_2}=2), but (\frac{c_1}{c_2}=\frac{9}{4}). Hence the lines are parallel and inconsistent.

Correct answer: C. (3x+y=10) and (x+4y=12)

In the third pair, (\frac{a_1}{a_2}\ne\frac{b_1}{b_2}). Hence the lines intersect at one point and the pair is consistent independent.

Correct answer: A. \(\left(0,-4\right)\) and \(\left(12,0\right)\)

At \(x=0\), \(y=-4\), and at \(y=0\), \(x=12\). While finding intercepts, note which variable is kept zero.

Correct answer: A. \(\left(\frac{9}{4},-\frac{3}{2}\right)\)

\(2.25=\frac{9}{4}\) and \(-1.5=-\frac{3}{2}\). It is better to convert decimal coordinates into simplified fractions.