Slope and y-intercept

In the line (y=mx+c), c is the y-intercept because x=0 on the y-axis. The first line has y-intercept b, while the second has y-intercept 4. For both lines to meet the y-axis at the same point, b=4. The terms 3x and -2x represent slopes, so they do not determine the y-intercept. Exam tip: put x=0 to find a y-intercept.

In the form \(y=mx+c\), the coefficient of \(x\) is the slope. Since the two lines have equal slopes, \(a=4\). Substituting \(x=2\) in the first line gives \(y=4\times2-5=3\). Option 4 is the value of the slope, not the value of \(y\). Exam tip: first equate the slopes to find the unknown coefficient, then substitute the given value of \(x\).

In the form y=-7x+c, the y-intercept is c. Since the y-intercept is 3, c=3. Substituting x=-2 gives y=-7(-2)+3=14+3=17. Hence, 17 is correct. The value 14 comes only from -7(-2); the y-intercept 3 must also be added. Exam tip: When the y-intercept is given, use it as the constant term c.

From \(3x-6=0\), we get \(x=2\). A line with a constant \(x\)-value is vertical, so it is parallel to the \(y\)-axis and has an undefined slope. Exam tip: whenever \(x=c\), identify the slope as undefined.

The slope of the line is \(-(p+1)\). Since an increase of 3 in \(x\) causes a decrease of 15 in \(y\), the slope is \(\frac{-15}{3}=-5\). Therefore, \(-(p+1)=-5\), so \(p+1=5\) and \(p=4\). Option 5 would result from incorrectly taking \(p+1\) as the answer. Exam tip: for change-based questions, find the slope using \(\frac{\Delta y}{\Delta x}\).

In \(y=mx+c\), \(m\) is the slope and \(c\) is the y-intercept. For \(y=-2x\), \(m=-2\) and \(c=0\), so it passes through the origin. Exam tip: check \(c=0\) first for an origin-passing line.

Given \(m+c=10\) and \(c=4\), substitute the value of \(c\): \(m+4=10\). Hence, \(m=6\). In the linear equation \(y=mx+c\), \(m\) represents the slope, so the correct answer is 6. The value 4 is only \(c\), the y-intercept, not the slope. Exam tip: in \(y=mx+c\), the coefficient of \(x\), \(m\), is the slope.

In the form y=mx+c, m is the slope and c is the y-intercept. Given m=-3, substitute it into m+c=5: -3+c=5, so c=8. Therefore, the y-intercept is 8. The value -8 can result from a sign error; moving -3 to the other side means adding 3. Exam tip: isolate c to find the y-intercept.

Move \(8x\) to the other side: \(4y=-8x+28\). Dividing by 4 gives \(y=-2x+7\). Comparing this with \(y=mx+c\), we get \(m=-2\) and \(c=7\). In option A, the y-intercept is correct, but the negative sign of the slope is missing. Exam tip: For \(Ax+By=C\), divide the whole equation by the coefficient of \(y\) after isolating the \(y\)-term.

The line x = 4 is parallel to the y-axis, so it never meets the y-axis and has no y-intercept. In contrast, y = 4 meets it at (0, 4). Exam tip: x = constant denotes a vertical line.

A line in the form \(y=mx+c\) has slope \(m\). Here, the coefficient of \(x\) is \(k+4\), so \(k+4=9\). Therefore, \(k=9-4=5\). Option \(9\) is the slope, not the value of \(k\). Exam tip: In \(y=mx+c\), identify the coefficient of \(x\) as the slope.

In \(y=mx+c\), \(m\) is the slope and \(c\) is the y-intercept. Option A has \(m=-2\), which is negative, and \(c=4\). Option C also has a negative slope but its intercept is \(-4\). Exam tip: check the coefficient of \(x\) and the constant term separately.

Substitute the point \((-3,14)\) in \(y=mx+5\): \(14=m(-3)+5\). Thus, \(14=-3m+5\), so \(-3m=9\) and \(m=-3\). Therefore, the slope is \(-3\). The number \(5\) is the y-intercept, not the slope. Exam tip: substitute the point's x- and y-coordinates in the line equation carefully.

In slope-intercept form, \(y\) is isolated; the coefficient of \(x\) is the slope and the constant is the y-intercept. In A, the slope is \(-3\) and the intercept is 7. Option B must first be rearranged for \(y\). Exam tip: look for \(y=mx+c\) directly.

In \(x=4\), the x-coordinate is fixed, so the graph is a vertical line and its slope is undefined. It does not meet the y-axis. Exam tip: an equation \(x=c\) always represents a vertical line.

In the form \(y=mx+c\), \(m\) is the slope and \(c\) is the y-intercept. For \(y=-2x+4\), the slope is \(-2\) and the intercept is 4. Exam tip: identify \(m\) and \(c\) directly first.

In the form \(y=mx+c\), the coefficient of \(x\) is the slope \(m\). Here, the slope is \(2p-3\), so \(2p-3=13\). Hence, \(2p=16\) and \(p=8\). If \(p=7\), the slope would be \(2(7)-3=11\), not 13. Exam tip: To identify the slope, look at the coefficient of \(x\).

In \(y=mx+c\), \(m\) is the slope and \(c\) is the y-intercept. For option A, \(m=-2\) gives a downward slope, while \(c=5>0\) places the y-intercept above the origin. Option C also has a negative slope, but its intercept is below the origin. Exam tip: check the signs of \(m\) and \(c\) separately.

In a linear model, the value of \(W\) at \(t=0\) is the initial amount, so it must be 120. The rate 8 litres per minute is the slope; hence \(W=8t+120\). Exam tip: substitute \(t=0\) to check the intercept.

To find the y-intercept, put x=0. Then -4y+12=0, so -4y=-12 and y=3. Hence, the line crosses the y-axis at (0,3), and its y-intercept is 3. The value -3 may result from a sign error. Exam tip: always set x=0 to find the y-intercept.