Slope and y-intercept

The slope-intercept form is \(y=mx+c\), where \(m\) is the slope and \(c\) is the y-intercept. With \(m=2\) and \(c=-3\), the equation is \(y=2x-3\). Option B has a negative slope. Exam tip: the coefficient of \(x\) gives the slope.

A line in the form \(y=mx+c\) has slope \(m\) and y-intercept \(c\). Here, \(m=3\) and \(c=-5\). Option C has the wrong sign for the intercept. Exam tip: identify the coefficient of \(x\) as the slope.

\(m\) represents the slope of the line. For a line parallel to the \(x\)-axis, the vertical change is \(\Delta y=0\), so \(m=\Delta y/\Delta x=0\). \(c=0\) only means that the line passes through the origin. Exam tip: a horizontal line always has zero slope.

To find the y-intercept, put x=0. Then 3(0)+4y=20, so 4y=20 and y=5. Hence, the line meets the y-axis at (0,5), and its y-intercept is 5. The number 20 is the constant term, not the y-intercept. Exam tip: always set x=0 to find a y-intercept.

Write the equation in slope-intercept form: \(6x+3y=21\Rightarrow 3y=-6x+21\Rightarrow y=-2x+7\). In the form \(y=mx+c\), the coefficient of \(x\), \(m\), is the slope. Therefore, the slope is \(-2\). Here, \(7\) is the y-intercept, not the slope. Exam tip: Convert the equation to \(y=mx+c\) first, then identify the coefficient of \(x\).

The slope of \(y=\frac{7}{3}x-4\) is \(\frac{7}{3}\), so for every 1-unit increase in \(x\), \(y\) increases by \(\frac{7}{3}\) units. Therefore, when \(x\) increases by 3, the increase in \(y\) is \(\frac{7}{3}\times 3=7\). The \(-4\) is the y-intercept and does not affect the change. Exam tip: multiply the slope by the change in \(x\) to find the change in \(y\).

The slope gives the change in y for each unit change in x. Here m = -5/6, and the increase in x is Δx = 12. Therefore Δy = mΔx = (-5/6)(12) = -10. The negative sign means y decreases, and the magnitude of that decrease is 10. Thus option C is correct. The intercept 11 does not affect the amount of change because it is a fixed constant; it only sets the starting vertical position. Options A and B arise from using only part of the fraction, while option D confuses the horizontal change with the vertical change.

The line is (y=mx+3) and it passes through the point ((4,23)). Substituting (x=4) and (y=23) gives (23=4m+3). Hence, (4m=20), so (m=5). If 4 were used instead, the value of (y) would be 19, not 23. Exam tip: To use a point on a line, substitute its (x)- and (y)-coordinates in the equation.

In the standard form \(y=mx+c\), \(m\) gives the slope. On putting \(x=0\), we get \(y=c\); hence \(c\) is the y-intercept. Exam tip: set \(x=0\) to identify the y-intercept.

The slope-intercept form of a line is \(y=mx+c\), where \(m\) is the slope and \(c\) is the y-intercept. Here, \(m=4\) and \(c=-7\), so the equation is \(y=4x-7\). Although \(y=4x+7\) has the correct slope, its y-intercept is \(+7\). Exam tip: From the point \((0,c)\), read the sign of the y-intercept carefully.

A line in slope-intercept form is \(y=mx+c\), where \(m\) is the slope and \(c\) is the \(y\)-intercept. Here, \(m=-6\), and \(y=5\) when \(x=0\), so \(c=5\). Therefore, the equation is \(y=-6x+5\). Option D has the correct slope but its \(y\)-intercept is \(-5\). Exam tip: the value of \(y\) at \(x=0\) is always the \(y\)-intercept.

In the form y=mx+c, m is the slope and c is the y-intercept. The coefficient of x is 5 in both equations, so both lines have slope 5. Their y-intercepts are 8 and -6 respectively, so they are not the same; lines with equal slopes and different y-intercepts are parallel and do not intersect. Exam tip: to identify the slope, look at the coefficient of x.

In the form y = mx + c, m is the slope and c is the y-intercept. The slopes of the two lines are 7 and -4, so they are not equal. Since the constant term in both equations is -3, both lines have the same y-intercept, -3. Exam tip: In y = mx + c, identify the slope from the coefficient of x and the y-intercept from 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. Substituting \(m=0\) and \(c=-8\) gives \(y=0x-8=-8\). In option A, the \(y\)-intercept is 0, while options C and D have non-zero slopes. Exam tip: A line with zero slope is horizontal and has the form \(y=\text{constant}\).

The slope-intercept form of a line is \(y=mx+c\), where \(m\) is the slope and \(c\) is the \(y\)-intercept. A line through the origin has \(c=0\). Since the slope is 11, its equation is \(y=11x\). Although \(y=11x+1\) also has slope 11, it does not pass through the origin because its \(y\)-intercept is 1. Exam tip: Substitute \(x=0\) to check whether the line passes through the origin.

The equation of the line is \(y=-x+10\). Substituting \(x=0\) gives \(y=-(0)+10=10\). Therefore, the correct answer is \(10\). The value \(-1\) is the coefficient of \(x\), not the value of \(y\) when \(x=0\). Exam tip: In \(y=mx+c\), putting \(x=0\) gives the y-intercept \(c\).

Substituting x=0 in the equation gives y=16-8(0)=16. Therefore, the value of y at x=0 is 16; this is also the y-intercept. The value -8 is the slope, or coefficient of x, not the value of y. Exam tip: To find the y-intercept, always put x=0.

To find the y-intercept, put x=0. Then y+6=4(0)=0, so y=-6. Hence, the line meets the y-axis at -6. Choosing 6 is a common sign error because +6 becomes -6 when moved to the other side. Exam tip: for a y-intercept, always set x=0.

Adding 8 to both sides of \(y-8=-5x\) gives \(y=-5x+8\). In slope-intercept form \(y=mx+c\), the coefficient of \(x\), namely \(m\), is the slope. Therefore, the slope is \(-5\). Here, \(8\) is the y-intercept, not the slope. Exam tip: Rewrite the equation in the form \(y=mx+c\) and identify the coefficient of \(x\).

The line is in the form \(y=ax-5\), where the coefficient of \(x\), namely \(a\), is the slope, while \(-5\) is the y-intercept. Since the slope is \(-7\), substituting \(a=-7\) gives \(y=-7x-5\). In option C, \(-7\) is the constant term, so its slope would be \(-5\). Exam tip: In \(y=mx+c\), the coefficient of \(x\) is always the slope \(m\).