Slope and y-intercept

In \(y=mx+c\), \(c\) is the y-intercept. Here \(c=4\), so the line meets the y-axis at \((0,4)\). The value \(-3\) is the slope, so it is not positive. Exam tip: put \(x=0\) to identify the y-intercept.

In \(y=5\), the coefficient of \(x\) is 0, so the slope is \(m=0\) and the line is horizontal. \(x=5\) is a vertical line. Exam tip: in \(y=mx+c\), \(m=0\) means the line is parallel to the x-axis.

In \(y=mx+c\), \(m\) is the slope. The given line has slope \(-3\), and option B also has slope \(-3\), so the lines are parallel. Option C has the same intercept, not the same slope. Exam tip: compare slopes for parallel lines.

To find the y-intercept, put \(x=0\). Substituting \(x=0\) in \(y=-2x-11\) gives \(y=-11\). Therefore, the line cuts the y-axis at \((0,-11)\). The point \((-11,0)\) lies on the x-axis, so it cannot be the y-intercept. Exam tip: every point on the y-axis has x-coordinate 0.

Divide both sides of \(4y=8x+12\) by \(4\) to get \(y=2x+3\). In the standard form \(y=mx+c\), \(m\) is the slope and \(c\) is the \(y\)-intercept. Therefore, the slope is \(2\) and the \(y\)-intercept is \(3\). Option A incorrectly uses the coefficients before dividing by \(4\). Exam tip: first rewrite the equation in the form \(y=mx+c\) before identifying the slope and intercept.

Write the line in slope-intercept form \(y=mx+c\) by dividing both sides by 5: \(y=-2x+5\). The coefficient of \(x\), \(m=-2\), is the slope, and the constant term, \(c=5\), is the \(y\)-intercept. Option A incorrectly uses the coefficients before dividing by 5. Exam tip: first rewrite the equation as \(y=mx+c\), then identify \(m\) and \(c\).

Rearranging \(x+y=6\) gives \(y=-x+6\). In \(y=mx+c\), the coefficient of \(x\) is the slope \(m\). Hence, the slope is \(-1\). Here, \(6\) is the y-intercept \(c\), not the slope. Exam tip: identify the coefficient of \(x\) after writing the equation in slope-intercept form.

To find the y-intercept, put x=0. In 2x+y=9, this gives y=9, so the y-intercept is 9; the corresponding point on the graph is (0, 9). Here, -2 is the slope because y=-2x+9. Exam tip: always set x=0 to find the y-intercept.

Write the equation in slope-intercept form: \(3x-y=6\Rightarrow -y=6-3x\Rightarrow y=3x-6\). In the form \(y=mx+c\), the coefficient \(m\) of \(x\) is the slope, so the slope is \(3\). The option \(-3\) is a common error because multiplying both sides by \(-1\) while isolating \(y\) makes the coefficient of \(x\) positive. Exam tip: convert the equation to \(y=mx+c\) before identifying the slope.

To find the y-intercept, put x=0. In 4x-y=-8, this gives -y=-8, so y=8. Therefore, the y-intercept is 8. The value -8 can result from a sign error, because both sides must change sign when solving -y=-8. Exam tip: always set x=0 to find the y-intercept.

The slope of \(y=\frac{3}{2}x-4\) is \(\frac{3}{2}\), so for every increase of 1 in \(x\), \(y\) increases by \(\frac{3}{2}\). Therefore, when \(x\) increases by 2, \(y\) increases by \(\frac{3}{2}\times2=3\). The constant \(-4\) is the y-intercept and does not affect the change. Exam tip: In \(y=mx+c\), use \(\Delta y=m\Delta x\) to find total change.

The slope of the line is \(-\frac{5}{3}\). This means that for every increase of 3 units in \(x\), \(y\) decreases by 5 units. Thus, \(\Delta y=-\frac{5}{3}\times 3=-5\), so \(y\) decreases by 5. In option B, 3 is the denominator of the slope, while 7 is only the y-intercept and does not affect the change. Exam tip: use \(\Delta y=m\Delta x\) to find the change in \(y\).

Substituting the point (2,7) into y=mx-1 gives 7=2m-1. Therefore, 2m=8 and m=4. If m=3, then y=2(3)-1=5, not 7. Exam tip: To find an unknown parameter in a line, substitute the coordinates of the given point into its equation.

In \(y=mx+c\), \(m\) is the slope and \(c\) is the y-intercept. In option A, \(m=-3\) is negative and \(c=4\). Option B has intercept 4 but a positive slope. Exam tip: check the coefficient of \(x\) for slope.

Putting \(x=0\) in \(y=2x+c\) gives \(y=c\). Since the line cuts the \(y\)-axis at \((0,5)\), \(c=5\). Therefore, the line is \(y=2x+5\). In \(y=5x+2\), the slope changes, whereas the given line has slope 2. Exam tip: in \(y=mx+c\), the \(y\)-intercept is \(c\).

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.