Slope and y-intercept

Correct answer: A. \(y=8x-11\)

The slope is \(m=\frac{\Delta y}{\Delta x}=\frac{40}{5}=8\). Since the line meets the (y)-axis at (0,-11), its (y)-intercept is \(-11\). Substituting \(m=8\) and \(c=-11\) in \(y=mx+c\) gives \(y=8x-11\). The equation \(y=40x-11\) has slope 40, while \(y=-8x-11\) represents a decreasing line. Exam tip: calculate slope as \(\Delta y/\Delta x\).

Correct answer: B. \(y=-7x+16\)

As \(x\) increases by \(6\), \(y\) decreases by \(42\), so the slope is \(m=\frac{-42}{6}=-7\). The line crosses the \(y\)-axis at \((0,16)\), hence its \(y\)-intercept is \(16\). Therefore, the equation is \(y=-7x+16\). In \(y=-42x+16\), the slope is \(-42\), not the required rate of change. Exam tip: find the slope using \(\frac{\Delta y}{\Delta x}\) first, then use the \(y\)-intercept.

Correct answer: C. \(19\)

The equation of the line is \(y=(a+7)x+(a-12)\). Since it passes through the origin \((0,0)\), substitute \(x=0\) and \(y=0\) to get \(a-12=0\), so \(a=12\). Hence, the slope, the coefficient of \(x\), is \(a+7=12+7=19\). Option \(18\) is a close distractor, but it does not result from substituting the correct value of \(a\). Exam tip: in \(y=mx+c\), \(m\) is the slope and \(c\) is the y-intercept.

Correct answer: B. 15

In the form y = mx + c, the y-intercept is the constant term c. Here, the y-intercept is u+5, so u+5=14 gives u=9. Therefore, the slope is 2u-3=2(9)-3=15. The value 13 is a distractor because it does not result from substituting the correct value of u into the slope expression. Exam tip: in y=mx+c, the coefficient of x is the slope and the constant term is the y-intercept.

Correct answer: A. (y)-intercept (10), slope (-\frac{8}{3})

The direct answer is option A: y-intercept 10 and slope \(-\frac{8}{3}\). Rewrite the equation in the form \(y=mx+c\). Starting with \(8x+3y-30=0\), move the other terms: \(3y=-8x+30\). Divide by 3: \(y=-\frac{8}{3}x+10\). Comparing with \(y=mx+c\), the slope is \(m=-\frac{8}{3}\), and the y-intercept is \(c=10\). Equivalently, putting \(x=0\) gives \(y=10\). Option A is therefore correct. Option B has both signs wrong. Option C treats the original coefficient 30 as the intercept and -8 as the slope without dividing by 3. Option D incorrectly uses the coefficients of x and y directly. The useful rule is: first isolate y; then the coefficient of x is the slope and the constant is the y-intercept.

Correct answer: C. 9

To find a line’s y-intercept, put x=0. The y-intercept of y=7x+b is b, while that of y=-5x+9 is 9. Since both lines cut the y-axis at the same point, their y-intercepts must be equal: b=9. The value 12 may come from the difference of the slopes, but slopes do not determine the y-intercept. Exam tip: for a y-intercept question, always substitute x=0.

Correct answer: B. 8

In the form y = mx + c, the coefficient of x, m, is the slope. Since the slopes are equal, a = 9. Thus, the first line is y = 9x - 10. Substituting x = 2 gives y = 9(2) - 10 = 18 - 10 = 8. Option 10 would result from not subtracting 10. Exam tip: compare the coefficients of x to compare slopes.

Correct answer: C. 26

A y-intercept of 6 means that y=6 when x=0, so c=6. Substituting x=-2 gives y=-10(-2)+6=20+6=26. Option 20 is only the value of -10(-2); the y-intercept 6 must still be added. Exam tip: multiplying a negative x-value by a negative coefficient gives a positive result.

Correct answer: C. 12

The slope of the line is \,\(k-4\). Therefore, when \,\(x\) increases by 7, the change in \,\(y\) is \,\(7(k-4)\). Given \,\(7(k-4)=56\), we get \,\(k-4=8\) and hence \,\(k=12\). If 11 were chosen, the slope would be 7, giving a change of only 49 in \,\(y\) for a change of 7 in \,\(x\). Exam tip: In change-based line questions, first write \,\(\text{slope}=\frac{\Delta y}{\Delta x}\).

Correct answer: C. 6

The slope of the line is \( -(p+3) \). When \(x\) increases by 4, the change in \(y\) is \(4[-(p+3)]\). Since \(y\) decreases by 36, \(4[-(p+3)]=-36\). Thus, \(p+3=9\), so \(p=6\). If \(p=5\), the slope would be \(-8\), giving a decrease of only 32. Exam tip: for a linear equation, use \(\Delta y=m\Delta x\) to find changes quickly.

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

\(x=3\) is a vertical line. Since \(\Delta x=0\), its slope is undefined, and it does not meet the y-axis \(x=0\). In contrast, \(y=3\) has zero slope. Exam tip: a line of the form \(x=\text{constant}\) is vertical.

Correct answer: C. 11

In the equation of a line, \(m\) represents the slope. Given \(m+c=18\) and \(c=7\), we get \(m+7=18\). Therefore, \(m=18-7=11\). Hence, 11 is the correct answer. The value 7 is \(c\), the y-intercept, not the slope. Exam tip: In \(y=mx+c\), identify \(m\) as the slope and \(c\) as the y-intercept.

Correct answer: C. 10

In the line equation y=mx+c, m is the slope and c is the y-intercept. Since the slope is -6, m=-6. Substituting in m+c=4 gives -6+c=4, so c=10. Therefore, the y-intercept is 10. The option -10 may result from a sign error. Exam tip: isolate c carefully when finding the y-intercept.

Correct answer: C. 7

Since \((10,15)\) lies on the line, substitute \(x=10\) and \(y=15\): \(15=\frac{4}{5}\times10+c\). Thus, \(15=8+c\), so \(c=7\). Option 8 could result from incorrectly evaluating \(\frac{4}{5}\times10\). Exam tip: To find an unknown constant in a line equation, substitute the coordinates of the given point.

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

In \(y=mx+c\), \(m\) is the slope and \(c\) is the y-intercept. A falling line needs \(m<0\), and an intercept 4 units below the origin means \(c=-4\); hence A is correct. Option B has the correct intercept but a positive slope. Exam tip: use the sign of \(m\) to identify the direction quickly.

Correct answer: A. 38

In a linear equation, the y-intercept is the constant term, the term not containing x. Here, the y-intercept is a+3. Thus, a+3=11 gives a=8. The slope is the coefficient of x: 5a-2=5(8)-2=38. Therefore, 38 is correct. Choosing 39 would give an incorrect coefficient of x. Exam tip: in y=mx+c, m is the slope and c is the y-intercept.

Correct answer: C. \(31\)

A line in the form \(y=mx+c\) has slope \(m\) and \(y\)-intercept \(c\). Here, the slope is \(b-11\). Given \(b-11=2\), we get \(b=13\). Therefore, the \(y\)-intercept is \(2b+5=2(13)+5=31\). Hence, option C is correct. \(29\) would result from incorrectly using \(b=12\). Exam tip: first equate the coefficient of \(x\) to the given slope, then substitute the parameter into the constant term.

Correct answer: A. 11

In the form \(y=mx+c\), \(m\) is the slope and \(c\) is the \(y\)-intercept. Equal slopes give \(t+4=12\), so \(t=8\). Therefore, the second line has \(y\)-intercept \(2t-5=2(8)-5=11\). Option 12 is the slope, not the \(y\)-intercept. Exam tip: first equate the slopes to find the parameter, then substitute it into the constant term.

Correct answer: C. 15

In the form \(y=mx+c\), \(c\) is the \(y\)-intercept. The first line has \(y\)-intercept \(v-7\), while the second has \(y\)-intercept \(5\). Therefore, \(v-7=5\), so \(v=12\). The slope of the second line is the coefficient of \(x\): \(v+3=12+3=15\). Option 14 is incorrect because it does not result after substituting the correct value of \(v\). Exam tip: in \(y=mx+c\), the coefficient of \(x\) is always the slope.

Correct answer: C. 24

The slope is \(-\frac{5}{3}\). Thus, for every increase of 3 units in \(x\), \(y\) decreases by 5 units. If the increase in \(x\) is \(\Delta x\), the decrease in \(y\) is \(\frac{5}{3}\Delta x\). Hence \(\frac{5}{3}\Delta x=40\), so \(\Delta x=40\times\frac{3}{5}=24\). Choosing 30 would decrease \(y\) by 50, not 40. Exam tip: a negative slope means that \(y\) decreases as \(x\) increases.

Correct answer: C. \(7\)

In this linear equation, the rate of change of \(y\) with respect to \(x\), or slope, is \(k+6\). Using \(\Delta y=(k+6)\Delta x\), we get \(52=(k+6)\times4\). Hence \(k+6=13\), so \(k=7\). If \(k=6\), the slope would be \(12\), giving an increase of only \(48\) in \(y\) for an increase of \(4\) in \(x\). Exam tip: the constant term \(+1\) does not affect the change in \(y\).

Correct answer: C. 9

In the line equation, the slope is represented by \(m\). Substituting \(c=-12\) into \(m-c=21\) gives \(m-(-12)=21\). Thus, \(m+12=21\), so \(m=9\). If 8 were chosen, then \(8-(-12)=20\), which does not satisfy the given condition of 21. Exam tip: subtracting a negative number is the same as adding it.

Correct answer: C. 11

Given m=-4 and 2c+m=18, substitute m to get 2c-4=18. Hence, 2c=22 and c=11. In the line y=mx+c, the y-intercept is c, so the correct answer is 11. If c were 12, then 2c+m would equal 20, not the given 18. Exam tip: in y=mx+c, identify the constant term c as the y-intercept.

Correct answer: D. 7

In the slope-intercept form \,y=mx+c\,, the coefficient of \,x\, is the slope. Here, the slope is \,3k-4\,. Therefore, \,3k-4=17\,. Adding 4 to both sides gives \,3k=21\,, and dividing by 3 gives \,k=7\,. If \,k=8\,, the slope would be \,20\,, not 17. Exam tip: In \,y=mx+c\,, identify the coefficient of \,x\, as the slope.

Correct answer: B. -9

Dividing the equation by 6 gives \(y=3x-9\). In the form \(y=mx+c\), the constant term \(c\) is the \(y\)-intercept. Therefore, the \(y\)-intercept is \(-9\), so the line passes through \((0,-9)\). Although \(-54\) is the constant term in the original equation, it must be divided by 6 to find the intercept. Exam tip: set \(x=0\) to find the \(y\)-intercept directly.