Slope and y-intercept

Both lines have the same slope \(m\), so they have the same direction. Since \(c\ne d\), their y-intercepts differ; therefore, they are not coincident but distinct parallel lines. Exam tip: same slope with different y-intercepts means parallel lines.

A y-intercept of 4 means that y=4 when x=0, so c=4. Substituting x=-3 gives y=-9(-3)+4=27+4=31. The value 27 is only the product -9×(-3); the intercept 4 must also be added. Exam tip: in y=mx+c, the y-intercept is the constant term c.

The slope of the line is \(k-3\). Since \(y\) increases by \(48\) when \(x\) increases by \(6\), the slope is \(\frac{48}{6}=8\). Thus, \(k-3=8\), so \(k=11\). If \(k=10\), the slope would be only \(7\), giving a rise of \(42\) for a run of \(6\). Exam tip: in \(y=mx+c\), \(m\) is the slope.

The slope of the line is \(-(p-1)\). An increase of \(5\) in \(x\) and a decrease of \(35\) in \(y\) give \(\Delta y=-35\). Hence, the slope is \(\frac{-35}{5}=-7\). Therefore, \(-(p-1)=-7\), so \(p-1=7\) and \(p=8\). If \(p=7\), the slope would be \(-6\), not the required \(-7\). Exam tip: Write a decrease in \(y\) as a negative change before finding the slope.

In \(y=mx+c\), \(c\) is the \(y\)-intercept and \(m\) is the slope. Thus, the same intercept requires \(c=5\), while a different slope requires \(m\ne-2\). Option B gives the same slope. Exam tip: do not interchange the roles of \(m\) and \(c\).

In the form (y=mx+c), m represents the slope. Given m+c=15 and c=6, we get m+6=15. Therefore, m=15-6=9. Hence, the correct answer is 9. Option 8 is incorrect because subtracting 6 from 15 gives 9. Exam tip: In y=mx+c, identify m as the slope and c as the y-intercept.

After shifting upward by 4 units, the equation becomes \(y=mx+(c+4)\). Thus, \(c\), the y-intercept, changes, while \(m\) remains unchanged because the line’s steepness is unchanged. Exam tip: a vertical shift changes only the constant term.

Since the given point \((8,10)\) lies on the line, substitute \(x=8\) and \(y=10\) in \(y=\frac{3}{4}x+c\). This gives \(10=\frac{3}{4}\times8+c=6+c\), so \(c=4\). Hence, 4 is the correct option. If \(c=3\), the right-hand side would be 9, not the given y-coordinate 10. Exam tip: For a line passing through a point, directly substitute the point’s coordinates into its equation.

In \(y=mx+c\), \(m\) is the slope and \(c\) is the y-intercept. Here \(m=-\frac{3}{4}\) and \(c=5\); putting \(x=0\) gives \(y=5\). Option D has slope \(-\frac{4}{3}\), not \(-\frac{3}{4}\). Exam tip: a negative slope means the line falls from left to right.

In the form y = mx + c, the y-intercept is c. Here, a + 2 = 9, so a = 7. Therefore, the slope is 4a - 1 = 4(7) - 1 = 27. Getting 28 usually results from an error in the subtraction step. Exam tip: first compare the constant term with the y-intercept to find the parameter, then use the coefficient of x to find the slope.

In a linear equation, the coefficient of x is the slope. Here the slope is b-9, so b-9=1 gives b=10. The y-intercept is the constant term 3b-2. Thus, 3(10)-2=28, so 28 is correct. Getting 30 would mean forgetting to subtract 2. Exam tip: put x=0 to find the y-intercept.

In the form \(y=mx+c\), \(m\) is the slope and \(c\) is the \(y\)-intercept. The slopes here are \(t+2\) and \(9\). Equal slopes give \(t+2=9\), so \(t=7\). The \(y\)-intercept of the second line is \(2t-1=2(7)-1=13\). Although \(11\) is the value of \(t+2\), it is not the \(y\)-intercept. Exam tip: find the parameter from the slopes first, then substitute it into the constant term.

In the form \(y=mx+c\), \(c\) is the \(y\)-intercept. The first line has \(y\)-intercept \(v-5\), while the second has \(y\)-intercept \(6\). Since they are equal, \(v-5=6\), so \(v=11\). The slope of the second line is the coefficient of \(x\), namely \(v+2\); hence it is \(11+2=13\). Option 11 is the value of \(v\), not the slope. Exam tip: in \(y=mx+c\), the coefficient of \(x\) is always the slope.

The slope is \(-\frac{4}{3}\). Thus, when \(x\) increases by 3 units, \(y\) decreases by 4 units. For a decrease of 28 in \(y\), \(\frac{4}{3}\Delta x=28\). Hence, \(\Delta x=28\times\frac{3}{4}=21\). Therefore, the correct answer is 21. An increase of 18 would decrease \(y\) by only 24. Exam tip: a negative slope means that \(y\) decreases as \(x\) increases.

In \(y=mx+c\), a falling line needs \(m<0\), while an intercept above the origin needs \(c>0\). Option A has \(m=-2\) and \(c=5\). Option C falls too, but its y-intercept is below zero. Exam tip: check the signs of \(m\) and \(c\) separately.

In the form (y=mx+c), m represents the slope. Given (m-c=16) and (c=-9), we get (m-(-9)=16), or (m+9=16). Hence, (m=7), so 7 is the correct option. If m were 8, then (m-c=8-(-9)=17), not 16. Exam tip: subtracting a negative value changes it to addition.

In the form (y=mx+c), the y-intercept is c. Substituting m=-3 into 3c+m=18 gives 3c-3=18. Hence, 3c=21 and c=7. Therefore, the y-intercept is 7. Option 6 may result from incorrectly ignoring the -3 term. Exam tip: to find the y-intercept, determine the value of c.