Slope and y-intercept

The slope-intercept form of a line is \(y=mx+c\), where \(m\) is the slope and \(c\) is the \(y\)-intercept. In \(y=3x+8\), \(m=3\) and \(c=8\), so option A is correct. Option B incorrectly interchanges the coefficient of \(x\) and the constant term. Exam tip: the coefficient of \(x\) gives the slope, while the constant term gives the \(y\)-intercept.

In \(y=mx+b\), \(m\) is the slope and \(b\) is the y-intercept. Parallel lines have equal slopes, so \(m=3\); with \(b=7\), the equation is \(y=3x+7\). Option A has the correct intercept but slope \(-3\). Exam tip: compare slope before intercept.

To find the slope, write the equation in the form \(y=mx+c\). Dividing \(2y=6x+10\) by 2 gives \(y=3x+5\). The coefficient of \(x\) is 3, so the slope is \(m=3\). Taking 6 as the slope is incorrect because it is the coefficient before making the coefficient of \(y\) equal to 1. Exam tip: after isolating \(y\), the coefficient of \(x\) is the slope.

To find the y-intercept, put x=0. Then 3y=9(0)-15=-15, so y=-5. Therefore, the line crosses the y-axis at -5. Taking -15 as the y-intercept is incorrect because it is the constant term before dividing the equation by 3. Exam tip: for the y-intercept, always substitute x=0.

The slope-intercept form is (y=mx+c), where (m) is the slope. The given equation can be written as (y=-2x+4), so the coefficient of (x) is -2 and the slope is -2. Although 2 is the magnitude, the negative sign shows that the line falls from left to right. Exam tip: In (y=mx+c), take the complete signed coefficient of (x) as the slope.

The line can be written in slope-intercept form as y=-3x+7. The y-intercept is the y-value when x=0. Substituting x=0 gives y=7, so the correct answer is 7. Here, -3 is the slope, not the y-intercept. Exam tip: In y=mx+c, c is 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. Substituting \(m=6\) and \(c=-4\) gives \(y=6x-4\). In option B, the intercept has the wrong sign, because its \(y\)-intercept is \(+4\). Exam tip: put \(x=0\); the constant term gives the \(y\)-intercept.

The slope-intercept form of a straight-line equation is \(y=mx+b\). Here, \(m\) represents the slope and \(b\) represents the \(y\)-intercept. The question gives slope \(-4\), so the coefficient of \(x\) must be \(-4\). It also gives a \(y\)-intercept of 9, so the constant term must be 9. Substituting these two values into the standard form identifies the equation.

Using \(m=-4\) and \(b=9\), we obtain \(y=-4x+9\). This is option D. The equation \(y=4x+9\) has the correct intercept but the wrong slope because its slope is positive. The other choices either interchange the two numbers or change their signs, so they do not satisfy both given properties.

For every point on the y-axis, x=0. Substituting x=0 in y=5x+c gives y=c. The given y-intercept is (0,-3), so c=-3. If c were 3, the line would meet the y-axis at (0,3), not at the given point. Exam tip: in y=mx+c, c is directly the y-intercept.

In the form \(y=mx+c\), \(m\) represents the slope. Since the given slope is \(-7\), \(m=-7\). The number \(2\) is the y-intercept, not the slope. Exam tip: in \(y=mx+c\), the coefficient of \(x\) is always the slope.

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 \(c\).