Slope and y-intercept

Correct answer: B. -7

Dividing 5y = 20x - 35 by 5 gives y = 4x - 7. In the form y = mx + c, the constant c is the y-intercept, so the y-intercept is -7. Here, 4 is the slope, not the y-intercept. Exam tip: You can also find the y-intercept by putting x = 0.

Correct answer: B. -4

Write the equation in slope-intercept form: \(4x+y=13\Rightarrow y=-4x+13\). In the form \(y=mx+c\), the coefficient of \(x\), namely \(m\), is the slope. Therefore, the slope is \(-4\). Here, 13 is the y-intercept, not the slope. Exam tip: Isolate \(y\) first to identify the slope quickly.

Correct answer: A. -18

To find the y-intercept, set \(x=0\). Substituting \(x=0\) in the equation gives \(-y=18\), so \(y=-18\). Hence, the y-intercept is \(-18\). The close distractor \(18\) results from overlooking the negative sign in \(-y=18\). Exam tip: always put \(x=0\) to find the y-intercept.

Correct answer: A. The line is parallel to the \(x\)-axis and intersects the \(y\)-axis at \(c\).

Putting \(m=0\) gives \(y=c\). Since the value of \(y\) remains constant, the line is parallel to the \(x\)-axis and meets the \(y\)-axis at \(c\). A line parallel to the \(y\)-axis has the form \(x=a\), not \(y=c\). Exam tip: \(m\) is slope and \(c\) is the y-intercept.

Correct answer: B. c

At every point on the \(y\)-axis, \(x=0\). Substituting \(x=0\) in \(y=mx+c\) gives \(y=c\), so \(c\) is the y-intercept. The term \(m\) represents the slope. Exam tip: set \(x=0\) to identify the y-intercept.

Correct answer: A. \(y=9x-4\)

The slope-intercept form of a line is \(y=mx+c\), where \(m\) is the slope and \(c\) is the \(y\)-intercept. Substituting \(m=9\) and \(c=-4\) gives \(y=9x-4\). In option B, the slope and the \(y\)-intercept have been interchanged. Exam tip: in \(y=mx+c\), the coefficient of \(x\) is always the slope.

Correct answer: C. \(y=-10x+8\)

The slope-intercept form of a line is \(y=mx+c\), where \(m\) is the slope and \(c\) is the \(y\)-intercept. Substituting \(m=-10\) and \(c=8\) gives \(y=-10x+8\). Option A has the wrong sign for the slope because its slope is \(10\). Exam tip: the coefficient of \(x\) gives the slope, while the constant term gives the \(y\)-intercept.

Correct answer: A. \(6\)

The slope-intercept form of a line is \(y=mx+c\), where \(m\) is the coefficient that represents the slope. Since the slope of \(y=mx-9\) is 6, \(m=6\). The value \(-9\) is the y-intercept, not the slope. Exam tip: In \(y=mx+c\), the coefficient of \(x\) is always the slope.

Correct answer: C. \(12\)

To find the \(y\)-intercept of \(y=-5x+c\), put \(x=0\). Then \(y=-5(0)+c=c\). Since the given point is \((0,12)\), \(c=12\). The value \(-5\) is the slope of the line, not its \(y\)-intercept. Exam tip: in \(y=mx+c\), \(c\) is the \(y\)-intercept directly.

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

The slope-intercept form is \(y=mx+c\), where \(m\) is the slope and \(c\) is the y-intercept. With \(m=2\) and \(c=-3\), the equation is \(y=2x-3\). Option B has a negative slope. Exam tip: the coefficient of \(x\) gives the slope.

Correct answer: A. The slope is 3 and the y-intercept is -5

A line in the form \(y=mx+c\) has slope \(m\) and y-intercept \(c\). Here, \(m=3\) and \(c=-5\). Option C has the wrong sign for the intercept. Exam tip: identify the coefficient of \(x\) as the slope.

Correct answer: A. \(m=0\)

\(m\) represents the slope of the line. For a line parallel to the \(x\)-axis, the vertical change is \(\Delta y=0\), so \(m=\Delta y/\Delta x=0\). \(c=0\) only means that the line passes through the origin. Exam tip: a horizontal line always has zero slope.

Correct answer: C. 5

To find the y-intercept, put x=0. Then 3(0)+4y=20, so 4y=20 and y=5. Hence, the line meets the y-axis at (0,5), and its y-intercept is 5. The number 20 is the constant term, not the y-intercept. Exam tip: always set x=0 to find a y-intercept.

Correct answer: B. \(-2\)

Write the equation in slope-intercept form: \(6x+3y=21\Rightarrow 3y=-6x+21\Rightarrow y=-2x+7\). In the form \(y=mx+c\), the coefficient of \(x\), \(m\), is the slope. Therefore, the slope is \(-2\). Here, \(7\) is the y-intercept, not the slope. Exam tip: Convert the equation to \(y=mx+c\) first, then identify the coefficient of \(x\).

Correct answer: B. 7

The slope of \(y=\frac{7}{3}x-4\) is \(\frac{7}{3}\), so for every 1-unit increase in \(x\), \(y\) increases by \(\frac{7}{3}\) units. Therefore, when \(x\) increases by 3, the increase in \(y\) is \(\frac{7}{3}\times 3=7\). The \(-4\) is the y-intercept and does not affect the change. Exam tip: multiply the slope by the change in \(x\) to find the change in \(y\).

Correct answer: C. 10

The governing concept is that the slope measures the change in y per unit change in x. In y = -(5/6)x + 11, the slope is m = -5/6. For an increase of Δx = 12, calculate the vertical change using Δy = mΔx: Δy = (-5/6)(12) = -10. The negative sign means that y decreases, and the amount of the decrease is the positive magnitude 10. Hence option C is correct. The constant 11 does not affect the change because it is added to every value of y and only determines the y-intercept. Options A and B result from mishandling the fraction, while option D confuses the horizontal change with the vertical change.

Correct answer: B. 5

The line is (y=mx+3) and it passes through the point ((4,23)). Substituting (x=4) and (y=23) gives (23=4m+3). Hence, (4m=20), so (m=5). If 4 were used instead, the value of (y) would be 19, not 23. Exam tip: To use a point on a line, substitute its (x)- and (y)-coordinates in the equation.

Correct answer: A. \(m\) slope and \(c\) y-intercept

In the standard form \(y=mx+c\), \(m\) gives the slope. On putting \(x=0\), we get \(y=c\); hence \(c\) is the y-intercept. Exam tip: set \(x=0\) to identify the y-intercept.

Correct answer: A. \(y=4x-7\)

The slope-intercept form of a line is \(y=mx+c\), where \(m\) is the slope and \(c\) is the y-intercept. Here, \(m=4\) and \(c=-7\), so the equation is \(y=4x-7\). Although \(y=4x+7\) has the correct slope, its y-intercept is \(+7\). Exam tip: From the point \((0,c)\), read the sign of the y-intercept carefully.

Correct answer: B. \(y=-6x+5\)

A line in slope-intercept form is \(y=mx+c\), where \(m\) is the slope and \(c\) is the \(y\)-intercept. Here, \(m=-6\), and \(y=5\) when \(x=0\), so \(c=5\). Therefore, the equation is \(y=-6x+5\). Option D has the correct slope but its \(y\)-intercept is \(-5\). Exam tip: the value of \(y\) at \(x=0\) is always the \(y\)-intercept.

Correct answer: A. Both have the same slope

In the form y=mx+c, m is the slope and c is the y-intercept. The coefficient of x is 5 in both equations, so both lines have slope 5. Their y-intercepts are 8 and -6 respectively, so they are not the same; lines with equal slopes and different y-intercepts are parallel and do not intersect. Exam tip: to identify the slope, look at the coefficient of x.

Correct answer: B. Both lines have the same y-intercept

In the form y = mx + c, m is the slope and c is the y-intercept. The slopes of the two lines are 7 and -4, so they are not equal. Since the constant term in both equations is -3, both lines have the same y-intercept, -3. Exam tip: In y = mx + c, identify the slope from the coefficient of x and the y-intercept from the constant term.

Correct answer: B. \(y=-8\)

The slope-intercept form of a line is \(y=mx+c\), where \(m\) is the slope and \(c\) is the \(y\)-intercept. Substituting \(m=0\) and \(c=-8\) gives \(y=0x-8=-8\). In option A, the \(y\)-intercept is 0, while options C and D have non-zero slopes. Exam tip: A line with zero slope is horizontal and has the form \(y=\text{constant}\).

Correct answer: A. \(y=11x\)

The slope-intercept form of a line is \(y=mx+c\), where \(m\) is the slope and \(c\) is the \(y\)-intercept. A line through the origin has \(c=0\). Since the slope is 11, its equation is \(y=11x\). Although \(y=11x+1\) also has slope 11, it does not pass through the origin because its \(y\)-intercept is 1. Exam tip: Substitute \(x=0\) to check whether the line passes through the origin.

Correct answer: A. \(10\)

The equation of the line is \(y=-x+10\). Substituting \(x=0\) gives \(y=-(0)+10=10\). Therefore, the correct answer is \(10\). The value \(-1\) is the coefficient of \(x\), not the value of \(y\) when \(x=0\). Exam tip: In \(y=mx+c\), putting \(x=0\) gives the y-intercept \(c\).