Slope and y-intercept

Correct answer: C. 8

In the form y=mx+c, m is the slope. Here, the coefficient of x is 8, so the slope is 8. Therefore, when x increases by 1, y increases by 8. The number 6 is the y-intercept, which gives the value of y when x=0; it is not the rate of increase. Exam tip: In y=mx+c, identify the coefficient of x to find the slope quickly.

Correct answer: D. (y) decreases by 6

The line is in the form y=mx+c, where m is the slope. Here, m=-6. Therefore, when x increases by 1, the change in y is -6, meaning y decreases by 6. The number 13 is the y-intercept; it does not give the change in y for a one-unit change in x. Exam tip: In y=mx+c, the change in y for an increase of 1 in x is exactly m.

Correct answer: C. -11

In y=7x-11, the term 7x depends on x, whereas -11 has no x. Therefore, the constant term is -11. The number 7 is the coefficient of x, not the constant term. Exam tip: A term with no variable is the constant term.

Correct answer: D. -9

In y = -9x + 16, the number multiplying x is -9, so the coefficient of x is -9. The number 16 is the constant term, not the coefficient of x. In this form, -9 also represents the slope of the line. Exam tip: the number that directly multiplies a variable is its coefficient.

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

In the form \(y=mx+c\), \(c\) is the \(y\)-intercept because putting \(x=0\) gives \(y=c\). In \(y=2x-3\), the constant term is \(-3\), so its \(y\)-intercept is \(-3\). In \(y=-3x+2\), \(-3\) is the slope, not the \(y\)-intercept. Exam tip: To find the \(y\)-intercept, put \(x=0\) in the equation.

Correct answer: C. \(y=6x-4\)

In the equation of a line \(y=mx+c\), the coefficient of \(x\), namely \(m\), is the slope. In \(y=6x-4\), the coefficient of \(x\) is 6, so its slope is 6. The slope of \(y=x+6\) is 1 because the coefficient of \(x\) is 1. Exam tip: identify the slope from the coefficient of \(x\), not from the constant term.

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

\(x=4\) is a vertical line, so its slope is undefined and it cannot fit \(y=mx+c\). In contrast, \(y=5\) has \(m=0\) and is a horizontal line. Exam tip: an equation \(x=\text{constant}\) always represents a vertical line.

Correct answer: A. Slope −8, y-intercept 5

A line in slope-intercept form is y=mx+c, where m is the slope and c is the y-intercept. The equation y=5−8x can be written as y=−8x+5. Therefore, m=−8 and c=5. In option C, the y-intercept is correct, but the sign of the slope is wrong. Exam tip: Read the coefficient of x carefully; the negative sign is part of the slope.

Correct answer: A. Slope

In the slope-intercept form y = mx + c, m represents the slope of the line, that is, the change in y for a one-unit change in x. Therefore, slope is correct. Here, -4 is the y-intercept and also the constant term, not m. Exam tip: In y = mx + c, remember m as the slope and c as the y-intercept.

Correct answer: C. y-intercept

In the form y=mx+c, m is the slope and c is the y-intercept. Here, putting x=0 gives y=c, so the line meets the y-axis at (0,c). The slope -3 describes the steepness and direction of the line, not its y-intercept. Exam tip: In y=mx+c, identify the constant term directly as the y-intercept.

Correct answer: D. \(-9\)

To find the y-intercept, put x=0. Then y=-0-9=-9, so the line meets the y-axis at \((0,-9)\) and its y-intercept is \(-9\). Here, \(-1\) is the slope, not the y-intercept. Exam tip: In y=mx+c, c is the y-intercept.

Correct answer: A. 15

The slope-intercept form of a line is y = mx + c, where m is the slope. In y = 15x + 2, the coefficient of x is 15, so the slope is 15. The number 2 is the y-intercept, not the slope. Exam tip: In y = mx + c, identify the coefficient of x as the slope.

Correct answer: A. \(0\)

In \(y=-12\), the value of \(y\) remains constant for every value of \(x\). Therefore, it represents a horizontal line parallel to the x-axis. Since the change in \(y\) is \(0\), its slope is \(m=\frac{\Delta y}{\Delta x}=0\). An undefined slope belongs to a vertical line, not a horizontal line. Exam tip: whenever you see \(y=\text{constant}\), the slope is \(0\).

Correct answer: C. 4

In the form (y=mx+c), (c) represents the y-intercept. The line meets the y-axis at (0,4), so the y-coordinate of the intercept is 4; hence (c=4). It would be -4 only if the intercept were (0,-4). Exam tip: the x-coordinate of every point on the y-axis is 0.

Correct answer: A. \(y=7x-1\)

The slope-intercept form of a line is \(y=mx+c\), where \(m\) is the slope and \(c\) is the \(y\)-intercept. Substituting \(m=7\) and \(c=-1\) gives \(y=7x-1\). Option C has slope 7, but its \(y\)-intercept is \(+1\), so it is not correct. Exam tip: in \(y=mx+c\), the coefficient of \(x\) is the slope and the constant term is the \(y\)-intercept.

Correct answer: B. \(y=-4x-6\)

The slope-intercept form of a line is \(y=mx+c\), where \(m\) is the slope and \(c\) is the \(y\)-intercept. Substituting \(m=-4\) and \(c=-6\) gives \(y=-4x-6\). Option D has the correct slope but its \(y\)-intercept is \(+6\). Exam tip: in \(y=mx+c\), the coefficient of \(x\) is the slope and the constant term is the \(y\)-intercept.

Correct answer: C. \(1\)

A line in the form \(y=mx+c\) has \(y\)-intercept \(c\). Here, \(c=1\), so the line meets the \(y\)-axis at \((0,1)\). \(\frac{7}{2}\) is the slope, not the \(y\)-intercept. Exam tip: put \(x=0\) to find the \(y\)-intercept.

Correct answer: B. \(-\frac{4}{7}\)

A line in slope-intercept form is \(y=mx+c\), where \(m\) is the slope and \(c\) is the y-intercept. In \(y=-\frac{4}{7}x+6\), the coefficient of \(x\) is \(-\frac{4}{7}\), so the slope is \(-\frac{4}{7}\). The number \(6\) is the y-intercept, not the slope. Exam tip: In \(y=mx+c\), always identify the coefficient of \(x\) as the slope.

Correct answer: C. 18

For every point on the y-axis, \(x=0\). Substituting \(x=0\) in \(y=6x+18\) gives \(y=6(0)+18=18\). Hence, the line cuts the y-axis at \(y=18\). The number \(6\) is the slope, not the y-intercept. Exam tip: in \(y=mx+c\), the y-intercept is \(c\).

Correct answer: C. Negative

In the form y=mx+c, the coefficient of x, m, is the slope. Here, m=-8, so the slope is negative. A zero slope occurs only for a horizontal line, whereas this line has slope -8. Exam tip: In y=mx+c, identify the slope by looking at the coefficient of x.

Correct answer: A. Positive

A line in the form y=mx+c has slope m, the coefficient of x. Here, m=12, and 12 is greater than 0; therefore, the slope is positive. A negative slope would require the coefficient of x to be less than 0. Exam tip: Identify the slope by looking at the coefficient of x in the equation.

Correct answer: A. Slope 0, y-intercept 3

The slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. In y = 0x + 3, the coefficient of x is m = 0 and the constant is b = 3. Therefore the line is horizontal and crosses the y-axis at (0, 3). Option A is correct. Options B and C exchange the two quantities, and option D changes the positive intercept to a negative one.

Correct answer: C. Coefficient -6, constant term 17

The expression can be written in standard form as (y=-6x+17). The number multiplying (x) is -6, so the coefficient of (x) is -6. The term independent of (x) is 17; hence, the constant term is 17. In option B, the sign of 6 is incorrect. Exam tip: Always include the positive or negative sign when identifying a coefficient.

Correct answer: B. y-अक्ष पर रेखा का अवरोध

In \(y=mx+c\), \(m\) is the slope, while \(c\) is the y-value when \(x=0\). Hence the line meets the y-axis at \((0,c)\). The x-intercept is different. Exam tip: put \(x=0\) to identify the y-intercept.

Correct answer: D. (y) decreases by 11

The slope of the line is -11. The slope gives the change in (y) for a 1-unit increase in (x). Therefore, when (x) increases by 1, (y) decreases by 11. The constant +4 is only the y-intercept; it does not give the rate of change. Exam tip: In y=mx+c, the change in y for a 1-unit increase in x is m.