Slope and y-intercept

The slope-intercept form of a line is \(y=mx+c\). Here, the slope is \(m=-3\) and the \(y\)-intercept is \(c=4\), so the equation is \(y=-3x+4\). Option A has the correct intercept but its slope is \(+3\), not \(-3\). Exam tip: from a point \((0,c)\) on the \(y\)-axis, the constant term is directly \(c\).

In the form y=mx+c, the coefficient m of x is the slope. Since the coefficient of x is 4 in both equations, both lines have slope 4 and are parallel. The values -6 and 9 are y-intercepts, not slopes. Exam tip: To find the slope, look for the coefficient of x.

In the form \(y=mx+c\), the y-intercept is \(c\), because \(x=0\) on the y-axis. Both equations have constant term \(5\), so both lines meet the y-axis at \(5\). The values \(-2\) and \(3\) are their slopes, not their y-intercepts. Exam tip: in \(y=mx+c\), \(m\) is the slope and \(c\) is the y-intercept.

A line can be written as \(y=mx+c\), where \(m\) is the slope and \(c\) is the y-intercept. Here, \(m=-6\) and \(c=0\), so the equation is \(y=-6x\). Although \(y=6x\) also has y-intercept 0, its slope is \(+6\). Exam tip: In \(y=mx+c\), the coefficient of x gives the slope.

The slope-intercept form of a line is \(y=mx+c\), where \(m\) is the slope and \(c\) is the \(y\)-intercept. Substituting \(m=0\) and \(c=9\) gives \(y=0x+9=9\). Hence, the line is horizontal and meets the \(y\)-axis at 9. Although \(y=x+9\) has a \(y\)-intercept of 9, its slope is 1. Exam tip: a horizontal line always has an equation of the form \(y=\) constant.

Substituting x=0 in the equation gives y=-(0)+6=6. Therefore, the correct value is 6. It is also the y-intercept because the line meets the y-axis where x=0. The value -1 is the coefficient of x (the slope), not the value of y. Exam tip: To find the y-intercept, always put x=0.

The given equation is (y=10-4x). Substituting (x=0) gives (y=10-4(0)=10). Hence, the correct answer is 10. Here, (-4) is the slope of the line, not the value of (y) when (x=0). Exam tip: In the form (y=mx+c), the value of (y) at (x=0) is always (c).

In the line y=mx+c, m is the slope and c is the y-intercept. Since m>0, the slope is positive; since c<0, the line cuts the y-axis below the origin, so the y-intercept is negative. Option B reverses both signs. Exam tip: In y=mx+c, identify m directly as the slope and c as the y-intercept.

In the equation y=mx+c, m is the slope and c is the y-intercept. A negative slope means that as x increases, y decreases, so the line falls from left to right. A positive c means that when x=0, y=c is above zero. Therefore the line crosses the y-axis at a positive point while having a decreasing direction.

Here m<0 gives a decreasing line, and c>0 gives a positive y-intercept. These two facts together match option B: decreasing and cutting the positive part of the y-axis. The line is not horizontal because its slope is not zero, and it does not necessarily pass through the origin because c is positive rather than zero.

Write the equation in slope-intercept form: \(y+5=2x\), so \(y=2x-5\). In the form \(y=mx+c\), the coefficient of \(x\), namely \(m\), is the slope. Therefore, the slope is \(2\). Here, \(-5\) is the y-intercept, not the slope. Exam tip: isolate \(y\) first and then identify the coefficient of \(x\).

Moving 7 to the right gives \(y=-3x+7\). To find the \(y\)-intercept, put \(x=0\); then \(y=7\). Hence, the \(y\)-intercept is \(7\). Here, \(-3\) is the slope, not the \(y\)-intercept. Exam tip: in \(y=mx+c\), \(c\) is the \(y\)-intercept.

The line is in the form \(y=ax+4\), where the coefficient of \(x\), namely \(a\), is the slope and 4 is the y-intercept. Since the slope is 9, substituting \(a=9\) gives \(y=9x+4\). In \(y=4x+9\), the slope is 4, so it is not correct. Exam tip: In \(y=mx+c\), the coefficient \(m\) of \(x\) is the slope.

In the form \(y=-2x+b\), the constant term \(b\) is the \(y\)-intercept. Since \(b=-6\), substituting it gives \(y=-2x-6\). Option C has the correct intercept but changes the coefficient of \(x\) from \(-2\) to \(+2\). Exam tip: in \(y=mx+c\), \(c\) is directly the \(y\)-intercept.

In the form \(y=mx+c\), \(m\) is the slope and \(c\) is the y-intercept. Both lines have \(m=3\), so they have the same slope. Their y-intercepts are \(1\) and \(-5\), respectively; hence, they are distinct parallel lines, not the same line. Exam tip: lines with equal slopes but different y-intercepts are parallel.

In the form y=mx+c, c is the y-intercept. In both equations, c=7, so both lines meet the y-axis at (0,7). Their slopes are 2 and -5, so the slopes are not the same; only the second line is decreasing. Exam tip: put x=0 to find the y-intercept.

In the line y=12, the value of y remains 12 at every point, so it is a horizontal line parallel to the x-axis. A horizontal line has slope 0, and it meets the y-axis at (0, 12); therefore, its y-intercept is 12. In option D, the slope is correct, but a y-intercept of -12 would represent the line y=-12. Exam tip: For every line of the form y=c, the slope is 0 and the y-intercept is c.

In the equation y = -9, the value of y remains constant for every value of x. Therefore, it is a horizontal line parallel to the x-axis, so its slope is 0. Putting x = 0 gives y = -9, hence the y-intercept is -9. Option B has the correct slope but the wrong sign for the y-intercept. Exam tip: every line of the form y = c has slope 0 and y-intercept c.

In \(y=mx+c\), \(m\) is the slope, which gives the change in \(y\) for each 1-unit change in \(x\). Here, the slope is \(\frac{1}{4}\). Therefore, when \(x\) increases by 8, \(\Delta y=\frac{1}{4}\times 8=2\). The number 6 is the y-intercept, not the change in \(y\). Exam tip: multiply the slope by the change in \(x\) to find the change in \(y\).

The slope is \(-\frac{2}{5}\), so for every increase of 5 units in \(x\), \(y\) decreases by 2 units. For an increase of 10 units, \(\Delta y=-\frac{2}{5}\times 10=-4\). Hence, \(y\) decreases by 4 units. Option 2 would apply only when \(x\) increases by 5 units. Exam tip: when asked “how much does it decrease,” report the magnitude of the negative change.

In slope-intercept form \(y=mx+c\), \(m\) is the slope and \(c\) is the y-intercept. With \(m=2\) and \(c=-3\), the equation is \(y=2x-3\). In \(y=2x+3\), the intercept has the wrong sign. Exam tip: check the coefficient of \(x\) and the constant separately.

In \(y=5\), the coefficient of \(x\) is 0, so the slope is 0. At \(x=0\), \(y=5\), giving y-intercept 5. \(x=5\) is a vertical line. Exam tip: identify \(m\) and \(c\) from \(y=mx+c\).

The slope-intercept form of a line is \(y=mx+c\), where \(m\) is the slope and \(c\) is the y-intercept. Here, \(m=3\), and \(y=4\) when \(x=0\), so \(c=4\). Therefore, the equation is \(y=3x+4\). In \(y=4x+3\), the slope is 4, so it is not correct. Exam tip: on putting \(x=0\), the resulting value of \(y\) is the constant term.

The slope-intercept form of a line is \(y=mx+c\), where \(m\) is the slope and \(c\) is the y-intercept. Here, \(m=-5\), and at \(x=0\), \(y=-2\); therefore, \(c=-2\). Hence, the line is \(y=-5x-2\). Option D has the correct slope but its y-intercept is \(+2\). Exam tip: Put \(x=0\) in an equation to identify its y-intercept directly.

A line written as \(y=mx+b\) can be understood by reading the coefficient of \(x\) as the slope and the constant term as the \(y\)-intercept. For \(y=8x-3\), the slope is 8 and the intercept is \(-3\). For \(y=-8x-3\), the slope is \(-8\) and the intercept is also \(-3\). Thus the lines have opposite slopes but meet the vertical axis at the same point.

Therefore their slopes are different, while their \(y\)-intercepts are the same, which is option B. They do not pass through the origin because putting \(x=0\) gives \(y=-3\), not zero. Their slopes are not zero either. The shared constant term is the key observation, while the opposite signs before 8 show that their directions differ.

In the form y=mx+c, m is the slope and c is the y-intercept. Both equations have -6 as the coefficient of x, so their slopes are the same. Their y-intercepts are 2 and -7 respectively, so they are different. Option D is wrong because -6 is a negative slope. Exam tip: In y=mx+c, the coefficient of x gives the slope and the constant term gives the y-intercept.