Slope and y-intercept

Correct answer: A. Slope \(3\) and \(y\)-intercept \(8\)

The slope-intercept form of a line is \(y=mx+c\), where \(m\) is the slope and \(c\) is the \(y\)-intercept. In \(y=3x+8\), \(m=3\) and \(c=8\), so option A is correct. Option B incorrectly interchanges the coefficient of \(x\) and the constant term. Exam tip: the coefficient of \(x\) gives the slope, while the constant term gives the \(y\)-intercept.

Correct answer: B. \(y=3x+7\)

In \(y=mx+b\), \(m\) is the slope and \(b\) is the y-intercept. Parallel lines have equal slopes, so \(m=3\); with \(b=7\), the equation is \(y=3x+7\). Option A has the correct intercept but slope \(-3\). Exam tip: compare slope before intercept.

Correct answer: C. 3

To find the slope, write the equation in the form \(y=mx+c\). Dividing \(2y=6x+10\) by 2 gives \(y=3x+5\). The coefficient of \(x\) is 3, so the slope is \(m=3\). Taking 6 as the slope is incorrect because it is the coefficient before making the coefficient of \(y\) equal to 1. Exam tip: after isolating \(y\), the coefficient of \(x\) is the slope.

Correct answer: B. -5

To find the y-intercept, put x=0. Then 3y=9(0)-15=-15, so y=-5. Therefore, the line crosses the y-axis at -5. Taking -15 as the y-intercept is incorrect because it is the constant term before dividing the equation by 3. Exam tip: for the y-intercept, always substitute x=0.

Correct answer: B. -2

The slope-intercept form is (y=mx+c), where (m) is the slope. The given equation can be written as (y=-2x+4), so the coefficient of (x) is -2 and the slope is -2. Although 2 is the magnitude, the negative sign shows that the line falls from left to right. Exam tip: In (y=mx+c), take the complete signed coefficient of (x) as the slope.

Correct answer: A. 7

The line can be written in slope-intercept form as y=-3x+7. The y-intercept is the y-value when x=0. Substituting x=0 gives y=7, so the correct answer is 7. Here, -3 is the slope, not the y-intercept. Exam tip: In y=mx+c, c is the y-intercept.

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

The slope-intercept form of a line is \(y=mx+c\), where \(m\) is the slope and \(c\) is the \(y\)-intercept. Substituting \(m=6\) and \(c=-4\) gives \(y=6x-4\). In option B, the intercept has the wrong sign, because its \(y\)-intercept is \(+4\). Exam tip: put \(x=0\); the constant term gives the \(y\)-intercept.

Correct answer: D. (y=-4x+9)

The slope-intercept form of a straight-line equation is \(y=mx+b\). Here, \(m\) represents the slope and \(b\) represents the \(y\)-intercept. The question gives slope \(-4\), so the coefficient of \(x\) must be \(-4\). It also gives a \(y\)-intercept of 9, so the constant term must be 9. Substituting these two values into the standard form identifies the equation.

Using \(m=-4\) and \(b=9\), we obtain \(y=-4x+9\). This is option D. The equation \(y=4x+9\) has the correct intercept but the wrong slope because its slope is positive. The other choices either interchange the two numbers or change their signs, so they do not satisfy both given properties.

Correct answer: B. -3

For every point on the y-axis, x=0. Substituting x=0 in y=5x+c gives y=c. The given y-intercept is (0,-3), so c=-3. If c were 3, the line would meet the y-axis at (0,3), not at the given point. Exam tip: in y=mx+c, c is directly the y-intercept.

Correct answer: C. -7

In the form \(y=mx+c\), \(m\) represents the slope. Since the given slope is \(-7\), \(m=-7\). The number \(2\) is the y-intercept, not the slope. Exam tip: in \(y=mx+c\), the coefficient of \(x\) is always the slope.

Correct answer: A. It intersects the y-axis at \((0,4)\)

In \(y=mx+c\), \(c\) is the y-intercept. Here \(c=4\), so the line meets the y-axis at \((0,4)\). The value \(-3\) is the slope, so it is not positive. Exam tip: put \(x=0\) to identify the y-intercept.

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

In \(y=5\), the coefficient of \(x\) is 0, so the slope is \(m=0\) and the line is horizontal. \(x=5\) is a vertical line. Exam tip: in \(y=mx+c\), \(m=0\) means the line is parallel to the x-axis.

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

In \(y=mx+c\), \(m\) is the slope. The given line has slope \(-3\), and option B also has slope \(-3\), so the lines are parallel. Option C has the same intercept, not the same slope. Exam tip: compare slopes for parallel lines.

Correct answer: B. (0,-11)

To find the y-intercept, put \(x=0\). Substituting \(x=0\) in \(y=-2x-11\) gives \(y=-11\). Therefore, the line cuts the y-axis at \((0,-11)\). The point \((-11,0)\) lies on the x-axis, so it cannot be the y-intercept. Exam tip: every point on the y-axis has x-coordinate 0.

Correct answer: B. Slope \(2\) and \(y\)-intercept \(3\)

Divide both sides of \(4y=8x+12\) by \(4\) to get \(y=2x+3\). In the standard form \(y=mx+c\), \(m\) is the slope and \(c\) is the \(y\)-intercept. Therefore, the slope is \(2\) and the \(y\)-intercept is \(3\). Option A incorrectly uses the coefficients before dividing by \(4\). Exam tip: first rewrite the equation in the form \(y=mx+c\) before identifying the slope and intercept.

Correct answer: C. Slope \,(-2) and \(y\)-intercept \,5

Write the line in slope-intercept form \(y=mx+c\) by dividing both sides by 5: \(y=-2x+5\). The coefficient of \(x\), \(m=-2\), is the slope, and the constant term, \(c=5\), is the \(y\)-intercept. Option A incorrectly uses the coefficients before dividing by 5. Exam tip: first rewrite the equation as \(y=mx+c\), then identify \(m\) and \(c\).

Correct answer: C. \(-1\)

Rearranging \(x+y=6\) gives \(y=-x+6\). In \(y=mx+c\), the coefficient of \(x\) is the slope \(m\). Hence, the slope is \(-1\). Here, \(6\) is the y-intercept \(c\), not the slope. Exam tip: identify the coefficient of \(x\) after writing the equation in slope-intercept form.

Correct answer: B. 9

To find the y-intercept, put x=0. In 2x+y=9, this gives y=9, so the y-intercept is 9; the corresponding point on the graph is (0, 9). Here, -2 is the slope because y=-2x+9. Exam tip: always set x=0 to find the y-intercept.

Correct answer: A. \(3\)

Write the equation in slope-intercept form: \(3x-y=6\Rightarrow -y=6-3x\Rightarrow y=3x-6\). In the form \(y=mx+c\), the coefficient \(m\) of \(x\) is the slope, so the slope is \(3\). The option \(-3\) is a common error because multiplying both sides by \(-1\) while isolating \(y\) makes the coefficient of \(x\) positive. Exam tip: convert the equation to \(y=mx+c\) before identifying the slope.

Correct answer: B. 8

To find the y-intercept, put x=0. In 4x-y=-8, this gives -y=-8, so y=8. Therefore, the y-intercept is 8. The value -8 can result from a sign error, because both sides must change sign when solving -y=-8. Exam tip: always set x=0 to find the y-intercept.

Correct answer: B. 3

The slope of \(y=\frac{3}{2}x-4\) is \(\frac{3}{2}\), so for every increase of 1 in \(x\), \(y\) increases by \(\frac{3}{2}\). Therefore, when \(x\) increases by 2, \(y\) increases by \(\frac{3}{2}\times2=3\). The constant \(-4\) is the y-intercept and does not affect the change. Exam tip: In \(y=mx+c\), use \(\Delta y=m\Delta x\) to find total change.

Correct answer: C. Decreases by 5

The slope of the line is \(-\frac{5}{3}\). This means that for every increase of 3 units in \(x\), \(y\) decreases by 5 units. Thus, \(\Delta y=-\frac{5}{3}\times 3=-5\), so \(y\) decreases by 5. In option B, 3 is the denominator of the slope, while 7 is only the y-intercept and does not affect the change. Exam tip: use \(\Delta y=m\Delta x\) to find the change in \(y\).

Correct answer: B. 4

Substituting the point (2,7) into y=mx-1 gives 7=2m-1. Therefore, 2m=8 and m=4. If m=3, then y=2(3)-1=5, not 7. Exam tip: To find an unknown parameter in a line, substitute the coordinates of the given point into its equation.

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

In \(y=mx+c\), \(m\) is the slope and \(c\) is the y-intercept. In option A, \(m=-3\) is negative and \(c=4\). Option B has intercept 4 but a positive slope. Exam tip: check the coefficient of \(x\) for slope.

Correct answer: A. \(y=2x+5\)

Putting \(x=0\) in \(y=2x+c\) gives \(y=c\). Since the line cuts the \(y\)-axis at \((0,5)\), \(c=5\). Therefore, the line is \(y=2x+5\). In \(y=5x+2\), the slope changes, whereas the given line has slope 2. Exam tip: in \(y=mx+c\), the \(y\)-intercept is \(c\).