Slope and y-intercept

Correct answer: A. \(2x-y=4\)

Rewriting \(2x-y=4\) as \(y=2x-4\) gives slope \(m=2\), which is positive, and y-intercept \(c=-4\), which is negative. Although \(-2x+y=4\) also has a positive slope, its intercept is \(4\). Exam tip: convert to \(y=mx+c\) before identifying signs.

Correct answer: A. \(3x-2y-8=0\)

The given line has slope \(3/2\). Option A gives \(y=(3/2)x-4\), so it is parallel and has intercept \(-4\). C has intercept \(-4\) but slope \(2/3\). Tip: compare \(m\) in \(y=mx+c\).

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

\(x=4\) is a vertical line, so its slope is undefined. It cannot meet the \(y\)-axis because every point on the \(y\)-axis has \(x=0\). In contrast, \(y=-3\) has slope 0. Exam tip: \(x=\text{constant}\) represents a vertical line.

Correct answer: D. 8

In the form \(y=mx+c\), the coefficient of \(x\) is the slope. Here, the slope is \(5p-2\), so \(5p-2=38\). Thus, \(5p=40\) and \(p=8\). If \(p=7\), the slope would be \(33\), so it is not correct. Exam tip: identify the coefficient of \(x\) to find the slope.

Correct answer: B. \(16\)

Since the line passes through the origin \((0,0)\), put \(x=0\) and \(y=0\). Its constant term must be zero: \(r-6=0\). Hence, \(r=6\). The slope is the coefficient of \(x\), so \(r+10=6+10=16\). Option \(6\) is the value of \(r\), not the slope. Exam tip: in \(y=mx+c\), \(m\) is the slope and \(c\) is the y-intercept.

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

Comparing with \(y=mx+c\), we get \(m=-3\) and \(c=4\). A negative slope means the line falls as we move right, and its \(y\)-intercept is 4. Exam tip: \(c\) always gives the \(y\)-axis intercept.

Correct answer: A. \(5\)

To find the y-intercept, put \(x=0\). This gives \(-6y+30=0\), so \(-6y=-30\) and \(y=5\). Hence, the y-intercept is \(5\). The value \(-5\) can result from a sign error. Exam tip: always set \(x=0\) to find the y-intercept.

Correct answer: C. 21

In the form \(y=mx+c\), the coefficient of \(x\) is the slope \(m\). The slopes of the first and second lines are \(q-9\) and \(12\), respectively. For equal slopes, \(q-9=12\), so \(q=21\). If 20 were chosen, the first slope would be 11, not 12. Exam tip: for parallel lines or lines with equal slopes, compare the coefficients of \(x\).

Correct answer: A. -19

In the form \(y=mx+c\), the constant term \(c\) is the \(y\)-intercept. The \(y\)-intercept of \(y=-10x+s\) is \(s\), while that of \(y=7x-19\) is \(-19\). Since the intercepts are equal, \(s=-19\). Note that \(-10\) is the slope of the first line, not its \(y\)-intercept. Exam tip: Put \(x=0\) to find the \(y\)-intercept.

Correct answer: A. It is true for every real value of a

For a line, the coefficient of x is its slope, so the slope is a+8. The constant term is the y-intercept, so the y-intercept is a. Their difference is (a+8)-a=8, which is true for every real value of a. Values such as a=8 or a=0 are only particular cases; the condition is not restricted to them. Exam tip: In y=mx+c, m is the slope and c is the y-intercept.

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

Parallel lines have equal slopes. The given line has slope \(-3\), while the required \(y\)-intercept is \(-2\); hence, in \(y=mx+c\), the equation is \(y=-3x-2\). Option D has the correct slope but intercept \(+2\). Exam tip: identify slope from the coefficient of \(x\) and intercept from the constant term.

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

In the form \(y=mx+c\), \(c\) is the y-intercept. For option A, \(y=2x-3\); putting \(x=0\) gives \(y=-3\). Hence, its y-intercept is \(-3\). In option C, \(-3\) is the slope, while option D can be written as \(y=\frac{x}{3}-\frac{2}{3}\). Exam tip: to find a y-intercept, put \(x=0\).

Correct answer: C. 16

In the form \(y=mx+c\), \(m\) is the slope. The first line has slope \(k-10\), while the second has slope \(6\). Since the slopes are equal, \(k-10=6\), so \(k=16\). For example, if \(k=15\), the first slope is 5, not 6. Exam tip: parallel lines have equal slopes; the constant term does not affect slope.

Correct answer: C. 5

In the form y = ax + b, the y-intercept is the constant term b. The first line has y-intercept -9, while the second has y-intercept m - 14. For equal y-intercepts, m - 14 = -9, so m = 5. If m were 4, the second y-intercept would be -10, so the intercepts would not match. Exam tip: put x = 0 to find the y-intercept.

Correct answer: A. \(y=-3x+5\)

In \(y=mx+c\), \(m\) is the slope and \(c\) is the y-intercept. Option A has \(m=-3\), so its slope is negative, and \(c=5\). Option B has intercept 5 but a positive slope. Exam tip: identify the slope from the coefficient of \(x\).

Correct answer: C. 16

The slope is \(\frac{9}{4}\), so an increase \(\Delta x\) in \(x\) produces an increase \(\frac{9}{4}\Delta x\) in \(y\). Thus, \(\frac{9}{4}\Delta x=36\), giving \(\Delta x=36\times\frac{4}{9}=16\). If \(x\) increased by 18, then \(y\) would increase by \(\frac{9}{4}\times18=40.5\), not 36. Exam tip: for change questions in \(y=mx+c\), use \(\Delta y=m\Delta x\); the constant term \(c\) does not affect the change.

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

In the form (y=px+q), p is the slope and q is the y-intercept. Here, p=-9, so y decreases as x increases; hence, the line decreases. Since q=14 is positive, the line meets the y-axis above the origin. Option C is incorrect because a horizontal line has slope 0. Exam tip: use the sign of the slope for increasing or decreasing behaviour, and use q for the y-intercept.

Correct answer: B. \(c_1=c_2\)

For a line \(y=mx+c\), setting \(x=0\) gives \(y=c\); therefore, \(c\) is its \(y\)-intercept. Hence the two lines have the same \(y\)-intercept if and only if \(c_1=c_2\). The condition \(m_1=m_2\) only gives equal slopes, so it does not guarantee equal \(y\)-intercepts. Exam tip: in \(y=mx+c\), \(m\) is the slope and \(c\) is the \(y\)-intercept.

Correct answer: C. 48

Write the equation in slope-intercept form: \(12y=-bx+96\), so \(y=-\frac{b}{12}x+8\). Hence, the slope is \(-\frac{b}{12}\). On setting \(-\frac{b}{12}=-4\), we get \(b=48\). If \(b=36\), the slope would be \(-3\), so it is not correct. Exam tip: for \(Ax+By=C\), the slope is \(-\frac{A}{B}\).

Correct answer: A. 0

The standard form of a line is \(y=mx+c\), where \(c\) is the \(y\)-intercept. Here, \(5n-7=38\), so \(5n=45\) and hence \(n=9\). Therefore, the slope is \(m=n-9=9-9=0\). Option 1 is not the slope because after finding \(n=9\), the expression \(n-9\) becomes zero. Exam tip: in \(y=mx+c\), the coefficient of \(x\) is the slope.

Correct answer: C. 69

A line in the form y = mx + c has slope equal to the coefficient of x. Thus, t - 12 = 6, so t = 18. The y-intercept is 4t - 3 = 4(18) - 3 = 69. Therefore, the correct answer is 69. The value 68 would result from incorrectly using 4t - 4 as the constant term. Exam tip: first find the parameter from the slope, then substitute it into the constant term to obtain the y-intercept.

Correct answer: A. \(y=10x-16\)

The slope is \(\frac{\Delta y}{\Delta x}=\frac{70}{7}=10\). Since the line meets the y-axis at \((0,-16)\), its y-intercept is \(-16\). Substituting \(m=10\) and \(c=-16\) in slope-intercept form \(y=mx+c\) gives \(y=10x-16\). In \(y=70x-16\), the slope would be 70, so a rise of 7 in x would increase y by 490. Exam tip: find the slope from the given changes first, then use the y-intercept.

Correct answer: B. \(y=-9x+20\)

When \(x\) increases by \(8\), \(y\) decreases by \(72\), so the slope is \(m=\frac{-72}{8}=-9\). The line meets the \(y\)-axis at \((0,20)\), so its \(y\)-intercept is \(20\). Therefore, the equation is \(y=-9x+20\). Option \(y=-72x+20\) has the correct intercept but an incorrect slope of \(-72\). Exam tip: find the slope from the change first, then use \(y=mx+c\) with the intercept.

Correct answer: C. 29

A line has the form y = mx + c, where c is the y-intercept. Since the line passes through the origin (0, 0), its y-intercept must be 0. Thus, a - 18 = 0, so a = 18. Therefore, the slope is m = a + 11 = 18 + 11 = 29. Option 28 would result from an error of 1 in finding a. Exam tip: for a line y = mx + c passing through the origin, always set c = 0.

Correct answer: B. 45

In the form y = mx + c, c is the y-intercept and m is the slope. Here, the y-intercept is u + 7. So, u + 7 = 19, giving u = 12. Therefore, the slope is 4u - 3 = 4(12) - 3 = 45. Option 47 may result from an arithmetic error in subtraction. Exam tip: first use the intercept to find the parameter, then substitute it into the slope expression.