Slope and y-intercept

Correct answer: C. 9

In the form y = mx + c, the coefficient of x, m, is the slope. Here, the slope is 2k - 5. So, 2k - 5 = 13, which gives 2k = 18 and k = 9. If k were 8, the slope would be 11, not 13. Exam tip: To identify slope, look for the coefficient of x.

Correct answer: B. -8

First divide the equation by 5: \(y=3x-8\). In the form \(y=mx+c\), the constant term \(c\) is the \(y\)-intercept. Therefore, the \(y\)-intercept is \(-8\), and the line meets the \(y\)-axis at \((0,-8)\). Although \(-40\) is the constant term in the original equation, it is not the intercept until the equation is written in slope-intercept form. Exam tip: set \(x=0\) to find the \(y\)-intercept directly.

Correct answer: C. \(9\)

Since the line \(y=-5x+c\) passes through \((4,-11)\), substitute \(x=4\) and \(y=-11\): \(-11=-5(4)+c=-20+c\). Hence, \(c=9\). Therefore, the y-coordinate of the \(y\)-intercept is \(9\), so the correct option is \(9\); the line meets the y-axis at \((0,9)\). Exam tip: in \(y=mx+c\), \(c\) directly gives the y-intercept value.

Correct answer: B. \(-5\)

Substitute the point \((-3,6)\) into the line: \(6=m(-3)-9\). Hence, \(6=-3m-9\), so \(15=-3m\) and \(m=-5\). Therefore, the slope is \(-5\). The value \(5\) may result from a sign error; the negative \(x\)-coordinate gives \(-3m\). Exam tip: substitute the given point carefully, keeping the signs intact, and then solve for the slope.

Correct answer: A. \(x=-4\)

In a line of the form \(x=a\), the x-coordinate remains fixed, so the line is vertical and parallel to the y-axis. \(y=4\) is horizontal instead. Exam tip: a constant \(x\) indicates a vertical line.

Correct answer: A. \(y=-2x+4\)

In \(y=mx+c\), \(m\) is the slope and \(c\) is the y-intercept. Option A has \(m=-2\), which is negative, and \(c=4\). Option B has a positive slope. Exam tip: check the coefficient of \(x\) first.

Correct answer: A. \(x=5\)

The line \(x=5\) is parallel to the y-axis. For a y-intercept, put \(x=0\), but \(0=5\) is impossible, so it never meets the y-axis. In contrast, \(y=5\) has y-intercept 5. Exam tip: \(x=a\), for \(a\ne0\), has no y-intercept.

Correct answer: C. 7

In the form \(y=mx+c\), the coefficient of \(x\) is the slope \(m\). Here, the slope is \(3p+1\). So, \(3p+1=22\), giving \(3p=21\) and \(p=7\). If \(p=6\), the slope would be \(3(6)+1=19\), not 22. Exam tip: identify the coefficient of \(x\) before equating slopes.

Correct answer: B. 9

The line is in the form y = mx + c, where the slope m = r + 6 and the y-intercept c = r - 3. A line passing through the origin (0, 0) must have y-intercept 0. Hence, r - 3 = 0, so r = 3. Therefore, the slope is r + 6 = 3 + 6 = 9. Option 3 is the value of r, not the slope. Exam tip: For y = mx + c to pass through the origin, set c = 0 directly.

Correct answer: A. The line goes down by 2 units for every 1 unit move to the right and cuts the y-axis at 6.

In y = mx + c, m is the slope and c is the y-intercept. Here m = -2, so y decreases by 2 when x increases by 1; c = 6. Exam tip: always check the sign of the slope.

Correct answer: A. 5

To find the y-intercept, put x=0 because every point on the y-axis has x-coordinate 0. Then -3y+15=0, so -3y=-15 and y=5. Hence, the y-intercept is 5. Taking -5 does not satisfy the equation. Exam tip: for a y-intercept, always set x=0.

Correct answer: D. 12

In the form y = mx + c, the coefficient of x, m, is the slope. The slopes of the first and second lines are q - 4 and 8 respectively. For equal slopes, q - 4 = 8, so q = 12. Option 8 is the slope of the second line, not the value of q. Exam tip: identify the coefficient of x when finding a slope.

Correct answer: A. \(-13\)

In the form \(y=mx+c\), \(c\) is the \(y\)-intercept. The \(y\)-intercept of \(y=-8x+s\) is \(s\), while that of \(y=5x-13\) is \(-13\). Since the intercepts are equal, \(s=-13\). Note that \(-8\) is the slope of the first line, not its intercept. Exam tip: put \(x=0\) to find the \(y\)-intercept.

Correct answer: A. It is true for every real value of a

Comparing the line with the standard form \(y=mx+c\), its slope is \(m=a+5\) and its \(y\)-intercept is \(c=a\). Therefore, their difference is \((a+5)-a=5\), which is true for every real value of a. Although it is true for \(a=5\) and \(a=0\), it is not restricted to those values. Exam tip: In \(y=mx+c\), the coefficient of x is the slope and the constant term is the \(y\)-intercept.

Correct answer: C. \(7\)

Substitute the point \((1,12)\) in \(y=5x+b\): \(12=5(1)+b\). Hence, \(b=12-5=7\). Checking with \(x=4\), we get \(y=5(4)+7=27\), so the second point also lies on the line. Taking \(b=5\) is incorrect because 5 is the slope, not the y-intercept. Exam tip: substitute the coordinates of a given point into the line equation to find an unknown constant.

Correct answer: A. \(y=-2x-3\)

In \(y=mx+c\), \(m=-2\) is negative, so the line falls from left to right. Since \(c=-3\), its y-intercept lies below the origin. Exam tip: check the signs of the slope and constant term separately.

Correct answer: C. 11

In the form y = mx + c, the coefficient of x, m, is the slope. The slopes of the two lines are k - 7 and 4 respectively. Since the slopes are equal, k - 7 = 4, so k = 11. If k were 12, the first slope would be 5, not 4. Exam tip: when comparing slopes, equate only the coefficients of x.

Correct answer: C. 4

In the form \(y=mx+c\), the y-intercept is the constant term \(c\). The first line has y-intercept \(-6\), while the second has y-intercept \(m-10\). For equal intercepts, \(m-10=-6\), so \(m=4\). If \(m=5\), the second intercept would be \(-5\), not \(-6\). Exam tip: put \(x=0\) to find a y-intercept.

Correct answer: A. रेखा बाएँ से दाएँ ऊपर जाती है या नीचे

\(m\) is the slope. If \(m>0\), the line rises from left to right; if \(m<0\), it falls. The y-intercept is determined by \(c\), not by the sign of \(m\). Exam tip: inspect the sign of \(m\) first.

Correct answer: C. 20

The slope is \(\frac{5}{4}\), so an increase of 4 units in \(x\) increases \(y\) by 5 units. Thus, \(\Delta y=\frac{5}{4}\Delta x\). Putting \(\Delta y=25\), we get \(25=\frac{5}{4}\Delta x\), hence \(\Delta x=25\times\frac{4}{5}=20\). Option 25 is the required change in \(y\), not the change in \(x\). Exam tip: for change-based questions on a line, use the slope; the constant term \(-9\) does not affect the change.

Correct answer: B. The line falls and the y-intercept is positive

The equation is in the form y = px + q, where p is the slope and q is the y-intercept. Since p = -7 is negative, the line goes downward as x increases, so it falls. Since q = 6 is positive, the line meets the y-axis at 6. Option C has the correct intercept but is wrong because a horizontal line must have slope 0. Exam tip: the sign of the slope tells whether a line rises or falls.

Correct answer: A. They are distinct parallel lines.

Both lines have the same slope, \(m\), so they are parallel. Since \(c_1\ne c_2\), their y-intercepts differ, so they cannot be the same line. Exam tip: equal slopes and different intercepts indicate distinct parallel lines.

Correct answer: A. \(x+2y+6=0\)

Writing the given line as \(y=2x+5\) gives slope \(2\). A perpendicular line has slope \(-\frac12\). With y-intercept \(-3\), it is \(y=-\frac12x-3\), or \(x+2y+6=0\). Exam tip: perpendicular slopes multiply to \(-1\).

Correct answer: A. 1

In the form \(y=mx+c\), the \(y\)-intercept is the constant term \(c\). Here, \(3n-2=19\), so \(3n=21\) and \(n=7\). Hence, the slope is \(n-6=7-6=1\). Option 2 could result from an incorrect calculation of \(n-6\). Exam tip: first use the constant term to find the parameter, then take the coefficient of \(x\) as the slope.

Correct answer: A. \(x=3\)

\(x=3\) is a vertical line parallel to the y-axis, \(x=0\), so it never meets the y-axis and has no y-intercept. In contrast, \(x+y=3\) gives \(y=3\) when \(x=0\). Exam tip: \(x=c\), for \(c\ne0\), has no y-intercept.