Slope and y-intercept

Correct answer: A. \(y=3x-5\)

In \(y=mx+c\), \(m>0\) means the line rises left to right, and \(c=-5\) puts its y-intercept below zero. Thus A is correct; B has \(m<0\). Exam tip: inspect \(m\) and \(c\) separately.

Correct answer: A. 15

In the form y = mx + c, the y-intercept is c. Here, the y-intercept is u + 3, so u + 3 = 11 gives u = 8. Therefore, the slope is 2u - 1 = 2(8) - 1 = 15. Option 16 can result from forgetting to subtract 1. Exam tip: first find the parameter using the y-intercept, then substitute it into the coefficient of x.

Correct answer: C. 6

To find a line’s y-intercept, put x=0. For the first line, y=b, while for the second line, y=6. Since both lines meet the y-axis at the same point, their y-intercepts must be equal. Hence, b=6. The number 5 is the slope of the first line, not its y-intercept. Exam tip: always set x=0 to find the y-intercept.

Correct answer: C. 11

In the form \(y=mx+c\), the coefficient of \(x\) is the slope. Since the slopes are equal, \(a=6\). Substituting \(x=3\) in the first line gives \(y=6\times3-7=11\). The \(5\) is the y-intercept of the second line, so it does not affect the y-value of the first line. Exam tip: first equate the slopes to find the unknown coefficient, then substitute the given value of \(x\).

Correct answer: D. 21

A y-intercept of 5 means that when x=0, y=5; therefore, c=5. The equation is y=-8x+5. Substituting x=-2 gives y=-8(-2)+5=16+5=21. Option 20 may seem close, but it does not result from adding the constant term 5 correctly. Exam tip: When substituting a negative value of x, check the sign of the product carefully.

Correct answer: A. \(y=-\frac{1}{2}x+3\)

The given line has slope \(2\). For perpendicular lines, the product of slopes is \(-1\), so the required slope is \(-\frac{1}{2}\). With y-intercept \(3\), the equation is \(y=-\frac{1}{2}x+3\). Option D has the correct slope but intercept \(-3\). Exam tip: check \(m\) and \(c\) separately in \(y=mx+c\).

Correct answer: A. \(y=40x+140\)

The slope is \((420-260)/(7-3)=40\) rupees per km. Substituting \(x=3, y=260\) in \(y=40x+b\) gives \(b=140\). Option B wrongly treats the charge at 3 km as the y-intercept. Exam tip: put \(x=0\) to check the fixed charge.

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

\(x=4\) is a vertical line. Its change in \(x\) is zero, so the slope \(\Delta y/\Delta x\) is undefined. Exam tip: any line of the form \(x=\text{constant}\) has an undefined slope.

Correct answer: C. \(7\)

Given \(m+c=12\) and \(c=5\), substituting the value of \(c\) gives \(m+5=12\). Hence, \(m=7\). In the equation \(y=mx+c\), \(m\) represents the slope, so the correct answer is \(7\). The value \(5\) is the y-intercept \(c\), not the slope. Exam tip: In \(y=mx+c\), the coefficient of \(x\), namely \(m\), is always the slope.

Correct answer: C. \(7\)

In the form \(y=mx+c\), \(m\) is the slope and \(c\) is the \(y\)-intercept. Given slope \(m=-4\). Substituting in \(m+c=3\) gives \(-4+c=3\), so \(c=7\). Choosing \(5\) would give \(m+c=1\), which does not satisfy the condition. Exam tip: In \(y=mx+c\), identify \(c\) directly as the \(y\)-intercept.

Correct answer: C. 5

Since the point \((6,9)\) lies on the line, substitute \(x=6\) and \(y=9\): \(9=\frac{2}{3}\times 6+c=4+c\). Hence, \(c=9-4=5\). Option B, 4, is only the value of \(\frac{2}{3}\times6\), not the y-intercept. Exam tip: To find an unknown constant in a line equation, substitute the coordinates of the given point.

Correct answer: C. \(y=-3x+4\)

The given line has slope \(\frac{1}{3}\). Slopes of perpendicular lines have product \(-1\), so the required slope is \(-3\). With y-intercept 4, the equation is \(y=-3x+4\). Exam tip: check the slope and constant term separately.

Correct answer: C. 13

In the form \(y=mx+c\), the \(y\)-intercept is the constant term \(c\). Here, \(a+1=6\), so \(a=5\). Therefore, the slope is \(m=3a-2=3(5)-2=13\). Option 12 would result from incorrectly ignoring the \(-2\). Exam tip: first use the \(y\)-intercept to find the parameter, then substitute it into the coefficient of \(x\).

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

Writing \(2x+4y=8\) as \(y=-\frac12x+2\) gives slope \(-\frac12\). A perpendicular line has slope \(2\); with y-intercept \(-3\), it is \(y=2x-3\). Exam tip: perpendicular slopes multiply to \(-1\).

Correct answer: C. \(10\)

The y-intercept of the first line is \(v-4\), while that of the second line is \(5\). Since the y-intercepts are equal, \(v-4=5\), so \(v=9\). The slope of the second line \(y=(v+1)x+5\) is \(v+1=9+1=10\). Note that \(9\) is the value of \(v\), not the slope. Exam tip: in \(y=mx+c\), \(m\) is the slope and \(c\) is the y-intercept.

Correct answer: C. 12

The slope is \(-\frac{3}{2}\). Thus, for every increase of 2 units in \(x\), \(y\) decreases by 3 units. If the increase in \(x\) is \(\Delta x\), the decrease in \(y\) is \(\frac{3}{2}\Delta x\). Hence, \(\frac{3}{2}\Delta x=18\), so \(\Delta x=12\). If 13 were chosen, the decrease would be \(\frac{3}{2}\times13=19.5\), not 18. Exam tip: With a negative slope, increasing \(x\) decreases \(y\).

Correct answer: A. Ravi is incorrect; the slope is 2 and the y-intercept is -3.

The slope is
\(m=(5-(-3))/(4-0)=2\). A y-intercept is the y-value when
\(x=0\), so it is -3 here. The value 5 is only the y-coordinate of
(4,5). Exam tip: first locate the point with
\(x=0\).

Correct answer: C. 8

Given \(c=-6\) and \(m-c=14\), substitute the value of \(c\): \(m-(-6)=14\), so \(m+6=14\). Hence, \(m=8\). Therefore, the slope of the line is 8. Choosing 6 is a common error; it is the magnitude of the negative value of \(c\), not the slope. Exam tip: always use brackets when substituting a negative value in \(m-c\).

Correct answer: C. 6

Given m=-2 and 2c+m=10, substituting m gives 2c-2=10. Hence, 2c=12 and c=6. In the line y=mx+c, the y-intercept is c, so the correct answer is 6. Choosing 5 is incorrect because it does not satisfy 2c-2=10. Exam tip: In y=mx+c, identify the constant term c to find the y-intercept.

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

In \(y=mx+c\), \(m\) is the slope and \(c\) is the y-intercept. In option A, \(m=2\) is positive and \(c=-3\) is negative. Option B has the correct intercept sign but a negative slope. Exam tip: check the signs of the coefficient of \(x\) and the constant separately.

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

In \(x=5\), x remains fixed, so the line is vertical, parallel to the y-axis, and has an undefined slope. \(y=5\) is horizontal with slope 0. Exam tip: \(x=\text{constant}\) represents a vertical line.

Correct answer: B. The line falls from left to right and intersects the y-axis at \(7\).

In \(y=mx+c\), \(m\) is the slope and \(c\) is the y-intercept. Here \(m=-3\), so the line falls as x increases, while \(c=7\) means it meets the y-axis at \(7\). Exam tip: a negative slope always indicates a downward line.

Correct answer: B. \(c_1=c_2\)

In \(y=mx+c\), putting \(x=0\) gives \(y=c\), so the y-axis point is \((0,c)\). Therefore, both lines have the same y-intercept only when \(c_1=c_2\). \(m_1=m_2\) indicates equal slopes, usually parallel lines. Exam tip: set \(x=0\) to identify a y-intercept.

Correct answer: B. 3

Substitute the point (-2,1) into y=mx+7: 1=m(-2)+7. Thus, 1=-2m+7, so -2m=-6 and m=3. Therefore, the slope is 3. If m were -3, the y-coordinate would be 13, not 1. Exam tip: substitute both coordinates carefully, especially when the x-coordinate is negative.

Correct answer: A. Its slope is undefined and it has no y-intercept.

In \(x=4\), the x-coordinate remains fixed, so the graph is a vertical line. Since \(\Delta x=0\), its slope is undefined, and it does not meet the y-axis. Exam tip: \(x=a\) always represents a vertical line.