Slope and y-intercept

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

In \(x=-4\), the x-coordinate is fixed, so the graph is a vertical line parallel to the y-axis. It cannot meet the y-axis, where \(x=0\). In contrast, \(y=-4\) is horizontal. Exam tip: \(x=a\) always represents a vertical line.

Correct answer: A. \(-3\)

In the form \(y=mx+c\), the constant term \(c\) is the \(y\)-intercept. The first line has \(y\)-intercept \(-4\), while the second has \(k-1\). For equal intercepts, \(k-1=-4\), so \(k=-3\). If \(-2\) were used, the second intercept would be \(-3\), not \(-4\). Exam tip: set \(x=0\) to identify a \(y\)-intercept.

Correct answer: C. 3

In the form \(y=mx+c\), the coefficient of \(x\) is the slope. The slopes of the first and second lines are \(m-1\) and \(2\), respectively. Equal slopes give \(m-1=2\), so \(m=3\). If 2 were chosen, the first slope would be 1, not equal to the second slope. Exam tip: identify the coefficient of \(x\) directly to find the slope.

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

In \(y=mx+c\), \(m\) is the slope and \(c\) is the y-intercept. In option A, \(m=-2\) is negative while \(c=5\) is positive. Option C has a negative slope too, but its intercept is negative. Exam tip: check the signs of the x-coefficient and constant separately.

Correct answer: B. 8

The slope is \(\frac{5}{2}\), meaning that an increase of 2 units in \(x\) increases \(y\) by 5 units. Hence, \(\Delta y=\frac{5}{2}\Delta x\). Putting \(\Delta y=20\), we get \(20=\frac{5}{2}\Delta x\), so \(\Delta x=20\times\frac{2}{5}=8\). If \(x\) increased by 10, \(y\) would increase by 25, not 20. Exam tip: For a line \(y=mx+c\), use \(\Delta y=m\Delta x\) in change-based questions.

Correct answer: B. The line decreases and the y-intercept is positive

In y=px+q, p is the slope and q is the y-intercept. Since p=-4 is negative, y decreases as x increases, so the line decreases. Since q=9 is positive, the line cuts the y-axis above the origin. Option C is incorrect because a horizontal line must have slope 0. Exam tip: use the sign of p to identify whether the line rises or falls, and the sign of q to locate the y-intercept.

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

In \(x=-3\), the x-coordinate stays fixed, so the graph is a vertical line parallel to the y-axis. It cannot meet the y-axis, where \(x=0\). In contrast, \(x=0\) is the y-axis itself. Exam tip: a constant-x line has undefined slope.

Correct answer: C. \(2x-3y=-12\)

Writing the given line as \(y=-\frac{3}{2}x+4\) gives slope \(-\frac{3}{2}\) and y-intercept 4. A perpendicular line has slope \(\frac{2}{3}\), so \(y=\frac{2}{3}x+4\), which is option C. Option B has intercept 4 but slope \(-\frac{2}{3}\). Exam tip: perpendicular slopes multiply to \(-1\).

Correct answer: A. 0

A line in the form y=mx+c has y-intercept c. Here, 2n+1=11, so 2n=10 and n=5. Therefore, the slope is n-5=5-5=0. Option 1 cannot be the slope because substituting n=5 makes the coefficient of x zero. Exam tip: In y=mx+c, the coefficient of x is always the slope.

Correct answer: A. \(-5\)

A line in the form \(y=mx+c\) has slope \(m\) and y-intercept \(c\). Here, the slope is \(r+4\). Thus, \(r+4=1\), giving \(r=-3\). Therefore, the y-intercept is \(r-2=-3-2=-5\). Note that \(-3\) is the value of \(r\), not the y-intercept. Exam tip: first find the parameter from the slope, then substitute it into the constant term.

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

An increase of \(10\) in \(y\) for an increase of \(2\) in \(x\) gives the slope \(m=\frac{10}{2}=5\). Since the line crosses the \(y\)-axis at \((0,-6)\), its \(y\)-intercept is \(b=-6\). Using \(y=mx+b\), the equation is \(y=5x-6\). The option \(y=10x-6\) has the correct intercept but an incorrect slope of \(10\). Exam tip: find the slope from the rate of change first, then identify the intercept by putting \(x=0\).

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

When \(x\) increases by 3 and \(y\) decreases by 12, the slope is \(m=\frac{-12}{3}=-4\). Since the line cuts the \(y\)-axis at \((0,9)\), its \(y\)-intercept is 9. Therefore, its equation is \(y=-4x+9\). Option A has a positive slope, but \(y\) is decreasing here. Exam tip: Use a negative sign for the slope whenever the dependent variable decreases.

Correct answer: A. The line falls from left to right and intersects the y-axis at \((0,4)\).

In \(y=mx+c\), \(m=-\frac{2}{3}\) is negative, so the line falls from left to right. Since \(c=4\), its y-intercept is \((0,4)\). Exam tip: use \(m\) for direction and \(c\) for the y-intercept.

Correct answer: A. 9

In the form y = mx + c, the y-intercept is the constant term c. Here, the y-intercept is t + 4. From t + 4 = 10, we get t = 6. Therefore, the slope is 2t - 3 = 2(6) - 3 = 9. Option 6 is the value of t, not the slope. Exam tip: set x = 0 to identify the y-intercept.

Correct answer: C. 4

In the line (y=mx+c), c is the y-intercept because x=0 on the y-axis. The first line has y-intercept b, while the second has y-intercept 4. For both lines to meet the y-axis at the same point, b=4. The terms 3x and -2x represent slopes, so they do not determine the y-intercept. Exam tip: put x=0 to find a y-intercept.

Correct answer: C. 3

In the form \(y=mx+c\), the coefficient of \(x\) is the slope. Since the two lines have equal slopes, \(a=4\). Substituting \(x=2\) in the first line gives \(y=4\times2-5=3\). Option 4 is the value of the slope, not the value of \(y\). Exam tip: first equate the slopes to find the unknown coefficient, then substitute the given value of \(x\).

Correct answer: C. 17

In the form y=-7x+c, the y-intercept is c. Since the y-intercept is 3, c=3. Substituting x=-2 gives y=-7(-2)+3=14+3=17. Hence, 17 is correct. The value 14 comes only from -7(-2); the y-intercept 3 must also be added. Exam tip: When the y-intercept is given, use it as the constant term c.

Correct answer: C. It is parallel to the \(y\)-axis and has an undefined slope.

From \(3x-6=0\), we get \(x=2\). A line with a constant \(x\)-value is vertical, so it is parallel to the \(y\)-axis and has an undefined slope. Exam tip: whenever \(x=c\), identify the slope as undefined.

Correct answer: B. 4

The slope of the line is \(-(p+1)\). Since an increase of 3 in \(x\) causes a decrease of 15 in \(y\), the slope is \(\frac{-15}{3}=-5\). Therefore, \(-(p+1)=-5\), so \(p+1=5\) and \(p=4\). Option 5 would result from incorrectly taking \(p+1\) as the answer. Exam tip: for change-based questions, find the slope using \(\frac{\Delta y}{\Delta x}\).

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

In \(y=mx+c\), \(m\) is the slope and \(c\) is the y-intercept. For \(y=-2x\), \(m=-2\) and \(c=0\), so it passes through the origin. Exam tip: check \(c=0\) first for an origin-passing line.

Correct answer: C. 6

Given \(m+c=10\) and \(c=4\), substitute the value of \(c\): \(m+4=10\). Hence, \(m=6\). In the linear equation \(y=mx+c\), \(m\) represents the slope, so the correct answer is 6. The value 4 is only \(c\), the y-intercept, not the slope. Exam tip: in \(y=mx+c\), the coefficient of \(x\), \(m\), is the slope.

Correct answer: C. 8

In the form y=mx+c, m is the slope and c is the y-intercept. Given m=-3, substitute it into m+c=5: -3+c=5, so c=8. Therefore, the y-intercept is 8. The value -8 can result from a sign error; moving -3 to the other side means adding 3. Exam tip: isolate c to find the y-intercept.

Correct answer: C. \(m=-2,\ c=7\)

Move \(8x\) to the other side: \(4y=-8x+28\). Dividing by 4 gives \(y=-2x+7\). Comparing this with \(y=mx+c\), we get \(m=-2\) and \(c=7\). In option A, the y-intercept is correct, but the negative sign of the slope is missing. Exam tip: For \(Ax+By=C\), divide the whole equation by the coefficient of \(y\) after isolating the \(y\)-term.

Correct answer: A. x = 4

The line x = 4 is parallel to the y-axis, so it never meets the y-axis and has no y-intercept. In contrast, y = 4 meets it at (0, 4). Exam tip: x = constant denotes a vertical line.

Correct answer: B. \(5\)

A line in the form \(y=mx+c\) has slope \(m\). Here, the coefficient of \(x\) is \(k+4\), so \(k+4=9\). Therefore, \(k=9-4=5\). Option \(9\) is the slope, not the value of \(k\). Exam tip: In \(y=mx+c\), identify the coefficient of \(x\) as the slope.