Slope and y-intercept

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.

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 \(x=0\).

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.

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.

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 \(x\).

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.

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.

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.

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 \(-\frac{A}{B}\).

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.

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.

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.

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.

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.

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.

From (5y=-12x+65), (y=-\frac{12}{5}x+13). So the (y)-intercept is (13) and slope is (-\frac{12}{5}).

In the form \(y=mx+c\), the \(y\)-intercept is \(c\), because \(x=0\) on the \(y\)-axis. The first line has \(y\)-intercept \(b\), while the second has \(y\)-intercept \(15\). Since they meet the \(y\)-axis at the same point, \(b=15\). The number \(9\) is the slope of the first line, not its intercept. Exam tip: in \(y=mx+c\), identify \(c\) directly as the \(y\)-intercept.

A line in the form \(y=mx+c\) has y-intercept \(c\). In option A, \(c=3\) for both lines, while the slopes are 2 and −5. Exam tip: compare the constant terms first to identify y-intercepts.

The y-intercept is the value of y when x=0. Therefore, in y=-12x+c, we have c=9. Substituting x=-4 gives y=-12(-4)+9=48+9=57. The value 48 is only the product -12×(-4); the y-intercept 9 must also be added. Exam tip: in y=mx+c, the y-intercept is c.

The slope of the line is (k-6). Since slope = change in y / change in x, (k-6)=108/9=12. Hence, (k=18). If 17 were chosen, the slope would be 11, so an increase of 9 in x would increase y by only 99, not 108. Exam tip: in a linear equation (y=mx+c), the constant c does not affect the rate of change; only m is the slope.