Slope and y-intercept

The slope of the line is \(-(p+5)\). A decrease of \(72\) in \(y\) for an increase of \(6\) in \(x\) gives slope \(\frac{-72}{6}=-12\). Therefore, \(-(p+5)=-12\), so \(p+5=12\) and \(p=7\). If \(p=8\), the slope would be \(-13\), giving a decrease of \(78\) for an increase of 6 in \(x\). Exam tip: In change-based questions, first find the slope using \(\frac{\Delta y}{\Delta x}\).

First isolate y to obtain the slope-intercept form. From 10y − 9x = 60, add 9x to both sides: 10y = 9x + 60. Dividing by 10 gives y = (9/10)x + 6. Comparing this with y = mx + b, the slope is m = 9/10 and the y-intercept is b = 6. Since x = 0 at the y-axis, substituting x = 0 gives y = 6. Therefore option A is correct. The negative answer in option B results from an incorrect sign change, while options C and D confuse original coefficients with the slope or intercept.

The y-intercept is where the line meets the y-axis, so \(x=0\). Substituting \(x=0\) in \(y=mx+c\) gives \(y=c\), hence the point is \((0,c)\). Do not confuse it with \((c,0)\), which is related to the x-axis. Exam tip: set \(x=0\) for a y-intercept.

In the form (y=mx+c), (m) is the slope and (c) is the y-intercept. Here, (m=-8). Substituting in (m+c=6) gives (-8+c=6), so (c=14). Therefore, the y-intercept is 14. Choosing (-14) would give (-8-14=-22), which does not satisfy the given condition. Exam tip: In (y=mx+c), the constant term (c) is the y-intercept.

Since \((16,25)\) lies on the line, substitute \(x=16\) and \(y=25\): \(25=\frac{7}{8}\times16+c=14+c\). Hence, \(c=25-14=11\). Option 10 can result from an error in multiplication or subtraction. Exam tip: for \(y=mx+c\), use \(c=y-mx\) after substituting the given point.

Writing the given line as \(y=\frac{2}{3}x+\frac{7}{3}\) shows that its slope is \(\frac{2}{3}\). A parallel line must have this same slope and y-intercept \(-4\). Option A rearranges to \(y=\frac{2}{3}x-4\). Exam tip: parallel lines always have equal slopes.

In the form y=mx+c, the y-intercept is c. Here, a+5=17, so a=12. Therefore, the slope is m=7a-6=7(12)-6=78. Option 76 would result if the expression were 7a-8, so it is not correct. Exam tip: first equate the constant term to the given y-intercept, then substitute the parameter into the coefficient of x.

In the form y = mx + c, the coefficient of x is the slope. Therefore, b - 15 = 5, so b = 20. The y-intercept is the constant term 4b + 2: 4(20) + 2 = 82. Hence, 82 is correct. The value 81 would result if the constant term were 4b + 1. Exam tip: first find the parameter using the slope, then substitute it into the constant term to get the y-intercept.

From (-10y=-9x-30), (y=\frac{9}{10}x+3). So the slope is (\frac{9}{10}) and the (y)-intercept is (3).

In the form \(y=mx+c\), \(m\) is the slope and \(c\) is the y-intercept. Since the slopes are equal, \(t+6=16\), so \(t=10\). For the second line, the y-intercept is \(2t-9=2(10)-9=11\). Therefore, 11 is correct. The value \(-10\) is the y-intercept of the first line, not the second. Exam tip: compare the coefficients of \(x\) for slopes and use the constant term for the y-intercept.

In the form \(y=mx+c\), the \(y\)-intercept is \(c\). The first line has \(y\)-intercept \(v-11\), while the second has \(y\)-intercept \(10\). Since they are equal, \(v-11=10\), so \(v=21\). Therefore, the slope of the second line is \(v+5=21+5=26\). Option \(25\) would result from incorrectly taking \(v=20\). Exam tip: in \(y=mx+c\), \(m\) is the slope and \(c\) is the \(y\)-intercept.

The slope is \(-\frac{9}{5}\), so for every increase of 5 units in \(x\), \(y\) decreases by 9 units. For a decrease of 54 in \(y\), \(\frac{9}{5}\Delta x=54\). Hence, \(\Delta x=54\times\frac{5}{9}=30\). If 35 were chosen, \(y\) would decrease by \(63\), not 54. Exam tip: in change-based questions, the constant 22 does not affect the change; use only the slope and \(\Delta x\).

In the form \(y=mx+c\), \(m\) is the slope and \(c\) is the y-intercept. Here \(m=-3\) and \(c=5\), so the line falls from left to right and meets the y-axis at \(5\). Exam tip: the coefficient of \(x\) gives the slope.

Given c=-18 and m-c=31, we get m-(-18)=31, or m+18=31. Hence, m=13. Therefore, the slope is 13. If m were 12, then m-c=12-(-18)=30, not 31. Exam tip: subtracting a negative number is the same as adding it.

In \(y=mx+c\), \(m\) is the slope and \(c\) is the y-intercept. Here \(m=3\) and \(c=-5\). Check by putting \(x=0\): \(y=-5\). Exam tip: set \(x=0\) to find the y-intercept.