Slope and y-intercept

Correct answer: A. 2

The slope-intercept form of a line is \(y=mx+c\), where \(m\) is the slope. In \(y=2x+5\), the coefficient of \(x\) is 2, so the slope is 2. The number 5 is the y-intercept, not the slope. Exam tip: In \(y=mx+c\), the coefficient of \(x\) gives the slope.

Correct answer: B. -4

In the form (y=mx+c), the constant term (c) is the (y)-intercept because substituting (x=0) gives (y=c). Here, in (y=3x-4), (c=-4), so the (y)-intercept is -4. The number 3 is the slope, not the (y)-intercept. Exam tip: To find the (y)-intercept, put (x=0) in the equation.

Correct answer: C. -1

The slope-intercept form of a line is y=mx+c, where m is the slope. The equation y=-x+7 can be written as y=(-1)x+7, so its slope is -1. Here, 7 is the y-intercept, not the slope. Exam tip: identify the coefficient of x and always retain its sign.

Correct answer: D. \(-2\)

The slope-intercept form of a line is \(y=mx+c\), where \(m\) is the slope. Writing the given equation as \(y=-2x+8\), the coefficient of \(x\) is \(-2\). Hence, the slope is \(-2\). Option 2 misses the negative sign, so it is not correct. Exam tip: Always check the sign of the coefficient of \(x\) while finding the slope.

Correct answer: A. 9

To find the y-intercept, put x=0. In y=x+9, this gives y=9, so the y-intercept is 9. Here, 1 is the slope of the line, not its y-intercept. Exam tip: in the form y=mx+c, c is the y-intercept.

Correct answer: B. 5

The standard form of a line is \(y=mx+c\), where \(m\) is the slope. In \(y=5x\), the coefficient of \(x\) is 5, so the slope is 5. A slope of 0 would represent a horizontal line, but this line rises as \(x\) increases. Exam tip: In \(y=mx+c\), the coefficient of \(x\) gives the slope directly.

Correct answer: C. -6

The line \(y=-6\) is a horizontal line. To find the \(y\)-intercept, put \(x=0\); the value of \(y\) is still \(-6\). Thus, the line meets the \(y\)-axis at \((0,-6)\), so its \(y\)-intercept is \(-6\). Option 6 is a close distractor, but the line has a negative value. Exam tip: for a line of the form \(y=c\), the \(y\)-intercept is always \(c\).

Correct answer: D. slope

In the equation y=mx+c, m is the coefficient of x and represents the slope of the line. It shows how much y changes when x increases by 1 unit. The value c is the y-intercept, so option A is not correct. Exam tip: in slope-intercept form, remember m as slope and c as y-intercept.

Correct answer: A. y-intercept

In the linear equation y=mx+c, putting x=0 gives y=c. Thus, c gives the point where the line crosses the y-axis, so it is the y-intercept. Here m is the slope, whereas the x-intercept is found by putting y=0. Exam tip: To find a y-intercept, always set x=0.

Correct answer: B. \(y=4x+1\)

The slope-intercept form of a line is \(y=mx+c\), where \(m\) is the slope and \(c\) is the \(y\)-intercept. In \(y=4x+1\), the coefficient of \(x\) is 4 and the constant term is 1, so it is the correct line. Although \(y=4x-1\) has slope 4, its \(y\)-intercept is \(-1\). Exam tip: In \(y=mx+c\), read the coefficient of \(x\) for slope and the constant term for the \(y\)-intercept.

Correct answer: C. y = -3x + 2

The governing concept is the slope-intercept form y = mx + c. In this form, m is the slope, which measures the change in y for a unit change in x, and c is the y-intercept, the value of y when x = 0. The required slope is -3, so the coefficient of x must be -3. The required y-intercept is 2, so the constant term must be +2. Substituting these values gives y = -3x + 2, which is option C. Option A has the correct intercept but a positive slope. Option B reverses the roles of the two numbers, making the slope 2 and the intercept -3. Option D has both an incorrect slope and an incorrect intercept, so it cannot represent the required line.

Correct answer: D. \(0\)

In the form \(y=mx+c\), \(c\) is the \(y\)-intercept. Here, in \(y=7x+0\), the constant term is \(0\), so the \(y\)-intercept is \(0\) and the line passes through the origin \((0,0)\). The number \(7\) is the slope, not the \(y\)-intercept. Exam tip: put \(x=0\) to find the \(y\)-intercept.

Correct answer: A. 0

The standard form of a line is \(y=mx+c\), where \(m\) is the slope. In \(y=0x+11\), the coefficient of \(x\) is 0, so the slope is 0. The number 11 is the y-intercept, not the slope. Exam tip: In \(y=mx+c\), identify the coefficient of \(x\) to find the slope.

Correct answer: B. -5

The slope-intercept form of a line is \(y=mx+c\), where \(m\) is the slope. In \(y=-5x-8\), the coefficient of \(x\) is \(-5\), so the slope is \(-5\). The number \(-8\) is the y-intercept, not the slope. Exam tip: In \(y=mx+c\), the coefficient of \(x\) gives the slope directly.

Correct answer: C. 12

The y-intercept is the y-coordinate where the line meets the y-axis, so x=0. Substituting x=0 in y=12+4x gives y=12. Therefore, the y-intercept is 12. The number 4 is the slope, not the y-intercept. Exam tip: in the form y=mx+c, c is the y-intercept.

Correct answer: D. 6

To find the y-intercept, put x=0 because every point on the y-axis has x-coordinate 0. Then y=6-3(0)=6, so the y-intercept is 6; the line meets the y-axis at (0,6). Here, -3 is the slope, not the y-intercept. Exam tip: In y=mx+c, c is the y-intercept.

Correct answer: A. Slope
(-2), (y)-intercept
(10)

The slope-intercept form of a line is
\(y=mx+c\), where
\(m\) is the slope and
\(c\) is the (y)-intercept. Comparing
\(y=-2x+10\) with this form gives
\(m=-2\) and
\(c=10\). Hence, the slope is
(-2) and the (y)-intercept is
(10). Option C has the correct intercept but misses the negative sign of the slope. Exam tip: always check the sign of the coefficient of
\(x\).

Correct answer: B. Slope 9, y-intercept −1

The slope-intercept form of a line is y = mx + c, where m is the slope and c is the y-intercept. In y = 9x − 1, the coefficient of x is 9, so the slope is 9; the constant term is −1, so the y-intercept is −1. Option C incorrectly interchanges these values. Exam tip: In y = mx + c, the coefficient of x is always the slope.

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

In \(y=4\), the y-coordinate remains constant while x can vary, so the graph is a horizontal line parallel to the x-axis. \(x=4\) is instead parallel to the y-axis. Exam tip: a constant y-value always gives a horizontal line.

Correct answer: B. 2

In the form \(y=mx+c\), \(c\) is the y-intercept. Here, \(c=2\), so the y-intercept is 2. \(-\frac{3}{4}\) is the slope of the line, not its y-intercept. Exam tip: set \(x=0\) to find the y-intercept.

Correct answer: B. (y) increases

A line with a positive slope rises from left to right. Therefore, as (x) increases, (y) generally increases. Option A describes a negative slope, while option D applies to a horizontal line with zero slope. Exam tip: a line that moves upward from left to right has a positive slope.

Correct answer: C. (y) decreases

A line with a negative slope goes downward as we move from left to right. Therefore, when (x) increases, the value of (y) decreases, so option C is correct. A constant value of (y) represents zero slope, not negative slope. Exam tip: If a graph falls as you move to the right, its slope is negative.

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

A line with slope 0 is horizontal and has the form \(y=c\), where \(c\) is a constant. Therefore, \(y=5\) has slope 0. The slopes of \(y=5x\), \(y=x+5\), and \(y=-5x\) are 5, 1, and -5 respectively. Exam tip: In \(y=mx+c\), the coefficient of \(x\), namely \(m\), is the slope.

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

In the form \(y=mx+c\), \(c\) is the y-intercept. A line passes through the origin \((0,0)\) only when \(c=0\). In \(y=4x\), the constant term is 0, so substituting \(x=0\) gives \(y=0\). The other options have constant terms 2, −5, and 1, so they do not pass through the origin. Exam tip: test for the origin by substituting \(x=0\) and \(y=0\) in the equation.

Correct answer: D. \((0,6)\)

Every point on the y-axis has x-coordinate 0. Substituting x=0 in y=2x+6 gives y=2(0)+6=6. Hence, the line cuts the y-axis at \((0,6)\). Although \((2,6)\) has y-coordinate 6, its x-coordinate is not 0, so it is not on the y-axis. Exam tip: To find the y-intercept, always put x=0.