Graphical method of finding solutions.

The x-intercept is the point where the line crosses the x-axis. Since every point on the x-axis has y-coordinate 0, put y = 0 in the equation x + y = 9. We obtain x + 0 = 9, so x = 9. Thus the x-intercept is (9, 0), and option A is correct. Option B is the y-intercept, because its x-coordinate is zero. Although (1, 8) and (4, 5) satisfy the equation, neither lies on the x-axis because their y-coordinates are not zero. This distinction is essential when reading a graph.

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

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.

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

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.

A single common point means that the pair has exactly one unique solution. Therefore, it is consistent and independent. In an inconsistent pair, the lines are parallel and have no solution, whereas a consistent dependent pair has coincident lines and infinitely many solutions. Exam tip: one point of intersection means one unique solution, so the pair is consistent and independent.

Coincident lines lie exactly on top of each other, so every point on the common line is a solution of both equations. Hence, the pair has infinitely many solutions and is called consistent and dependent. Option C is incorrect because an independent pair has lines that intersect at exactly one point. Exam tip: coincident lines imply infinitely many solutions and a consistent dependent pair.

For the ordered pair \((7, 3)\), \(x+y=7+3=10\) and \(x-y=7-3=4\). Thus, it satisfies both equations, so \((7,3)\) is the intersection point of the two lines on the graph. For example, \((6,4)\) satisfies the first equation but \(6-4=2\), not 4. Exam tip: substitute the coordinates into both equations to verify a solution.

Subtracting the second equation from the first gives \\(x=2\\). Substituting this into \\(x+y=5\\) gives \\(y=3\\). Therefore, the intersection point is \\(2,3\\). For example, option B gives \\(2(3)+2=8\\), so it does not satisfy the first equation. Exam tip: verify a graphical-solution point in both equations.

A solution of two linear equations is an ordered pair whose first number is the value of x and whose second number is the value of y. The same pair must make both equations true at the same time. This is also the point where the two corresponding lines intersect on a graph.

For option A, take x=3 and y=2. In the first equation, x+3y becomes 3+3(2)=3+6=9, so it is satisfied. In the second equation, x+y becomes 3+2=5, so that equation is also satisfied. Therefore, the common solution is (3,2). A pair that satisfies only one equation would not be a solution of the pair.

Both axes meet at ( (0,0) ). While reading a graph, it is easy to count coordinates from the origin.

To find the x-intercept, put \(y=0\), giving \(x=6\) and the point \((6,0)\). To find the y-intercept, put \(x=0\), giving \(y=6\) and the point \((0,6)\). Hence, A is correct. In option B, both coordinates are 3; although \(3+3=6\), those points are not on the respective axes. Exam tip: set \(y=0\) for the x-intercept and \(x=0\) for the y-intercept.

Putting \(x=0\) in \(2x+y=4\) gives \(y=4\), so \((0,4)\) lies on the line. Putting \(y=0\) gives \(2x=4\), hence \(x=2\), so \((2,0)\) is the second point. The points in option B do not satisfy the equation; for example, substituting \((4,0)\) gives a left side of 8. Exam tip: To obtain two convenient points, set \(x=0\) and \(y=0\) successively.

Substituting the coordinates of \((4,2)\) gives \(x-2y=4-2(2)=4-4=0\). Therefore, the point lies on the line. The other points do not make the left-hand side equal to zero. In such questions, verify a point by substituting its coordinates directly into the equation.

In an ordered pair \((x, y)\), the first value is the \(x\)-coordinate and the second value is the \(y\)-coordinate. Therefore, for \((1, 2)\), \(x=1\). Option 2 is the \(y\)-coordinate, not the \(x\)-coordinate. Exam tip: Always read coordinates in the order \((x, y)\).

In an ordered pair (x, y), the first coordinate is x and the second coordinate is y. Therefore, the y-coordinate of (5, 2) is 2. Option 5 is the x-coordinate, not the y-coordinate. Exam tip: Always read the coordinates in the order (x, y).

The equation \(x=0\) means that the \(x\)-coordinate of every point is zero, while the \(y\)-coordinate may have any value. All such points lie on the \(y\)-axis, so the correct answer is the \(y\)-axis. The \(x\)-axis is represented by \(y=0\), which is a useful exam tip to remember.

The equation \(y=0\) means that the y-coordinate of every point is zero. All such points lie on the x-axis, so this is the equation of the x-axis. The y-axis is represented by \(x=0\). Exam tip: a horizontal line has a constant y-value, while a vertical line has a constant x-value.

Both lines are parallel to the (y)-axis and have different (x)-values. Therefore, they are parallel and will not meet.

The equations \(y=1\) and \(y=4\) represent horizontal lines. Since their \(y\)-intercepts are different, they never meet and are parallel. They are not coincident because coincident lines have all points in common. Exam tip: the graph of \(y=c\) is always a horizontal line.