Slope and y-intercept

In \(y=3x-4\), putting \(x=0\) gives \(y=-4\), so the graph meets the y-axis at \((0,-4)\). Option C has y-intercept \(-4/3\). Exam tip: set \(x=0\) to identify the y-intercept.

In \(y=mx+c\), \(m\) represents the slope. A line parallel to the \(x\)-axis is horizontal, so its slope is \(0\). The condition \(c=0\) only means that the line passes through the origin. Exam tip: for a line parallel to the \(x\)-axis, always use \(m=0\).

In the form y = mx + c, the coefficient of x, m, is the slope. Here, the slope is 3p - 2. So, 3p - 2 = 16, which gives 3p = 18 and p = 6. Therefore, option C is correct. If p were 5, the slope would be 13, not 16. Exam tip: In y = mx + c, identify the coefficient of x to find the slope.

The line is \(y=(r-4)x+(r+6)\). Since it passes through the origin \((0,0)\), substituting \(x=0\) and \(y=0\) gives \(r+6=0\). Hence, \(r=-6\). Therefore, the slope is \(r-4=-6-4=-10\). Note that \(-6\) is the value of the parameter \(r\), not the slope. Exam tip: in \(y=mx+c\), \(m\) is the slope and \(c\) is the y-intercept.

Rearranging the given equation gives 3y=-12x+21. Dividing by 3, we get y=-4x+7. In the form y=mx+c, m is the slope and c is the y-intercept. Hence, the slope is -4 and the y-intercept is 7. A common close error is to use a positive slope; moving 12x to the other side makes its sign negative. Exam tip: Rewrite the line in y=mx+c form before identifying the slope and intercept.

To find the y-intercept, put x=0 because every point on the y-axis has x-coordinate 0. This gives -2y+10=0, so -2y=-10 and y=5. Hence, the line meets the y-axis at (0,5), and its y-intercept is 5. The value -4 can result from a sign error. Exam tip: always set x=0 to find the y-intercept.

In the form \(y=mx+c\), the coefficient of \(x\) is the slope. The slopes of the first and second lines are \(q+3\) and \(11\), respectively. For equal slopes, \(q+3=11\), so \(q=8\). If \(q=7\), the slope would be \(10\), not \(11\). Exam tip: while comparing slopes, use the coefficient of \(x\), not the constant terms such as \(-6\) or \(2\).

In the line equation \(y=mx+c\), the constant term \(c\) is the y-intercept. The y-intercept of \(y=-7x+s\) is \(s\), while that of \(y=4x-12\) is \(-12\). Since the intercepts are equal, \(s=-12\). Note that \(-7\) is the slope of the first line, not its y-intercept. Exam tip: put \(x=0\) to find the y-intercept.

In the standard form y=mx+c, m is the slope and c is the y-intercept. Here, the slope is a-4 and the y-intercept is a. Hence, their difference is \((a-4)-a=-4\), which does not depend on the value of a. Therefore, the statement is true for every real value of a. Although a=4 also works, it is not the only value. Exam tip: Identify m and c first, then simplify the stated difference.

The slope is \(m=\frac{-7-5}{4-(-2)}=-2\). Using \((-2,5)\), \(5=4+b\), so \(b=1\). The value 5 is not a y-intercept because its x-coordinate is -2. Exam tip: put \(x=0\) to find the y-intercept.

Writing \(3x+4y=12\) as \(y=-\frac{3}{4}x+3\) shows that the coefficient of \(x\) is the slope and the constant is the y-intercept. Option A misses the negative sign. Exam tip: isolate \(y\) first.

In the form y = mx + c, the coefficient of x, m, is the slope. The first line has slope k - 6 and the second has slope 5. For equal slopes, k - 6 = 5, so k = 11. If k were 10, the first slope would be 4, so it would not match. Exam tip: identify the coefficient of x for the slope; the constant term does not affect it.

In the form \(y=mx+c\), the \(y\)-intercept is the constant term \(c\). The first line has \(y\)-intercept \(-5\), while the second has \(y\)-intercept \(m-9\). For equal intercepts, \(m-9=-5\), so \(m=4\). If \(m=5\), the second intercept would be \(-4\), not \(-5\). Exam tip: put \(x=0\) to find a \(y\)-intercept.

For A, \(2y=-3x+6\), so \(y=-\frac{3}{2}x+3\). Its slope is negative and its y-intercept is 3, so it crosses the y-axis above the origin. C also has a negative slope, but its intercept is \(-3\). Exam tip: first rewrite the equation as \(y=mx+c\).

The slope is \(\frac{4}{3}\), meaning that for every increase of 3 units in \(x\), \(y\) increases by 4 units. Thus, \(\Delta y=\frac{4}{3}\Delta x\). Putting \(\Delta y=24\), we get \(24=\frac{4}{3}\Delta x\), so \(\Delta x=24\times\frac{3}{4}=18\). If \(x\) increased by 24, then \(y\) would increase by 32, not 24. Exam tip: In change-based questions, the constant term \(-6\) does not affect the change; use only the slope.

In (y=px+q), p is the slope of the line and q is its y-intercept. Here, (p=6) is positive, so y increases as x increases and the line rises. Also, (q=-8) means that the line cuts the y-axis at -8, so its y-intercept is negative. Option C is incorrect because a horizontal line has slope 0. Exam tip: the sign of the slope shows whether a line rises or falls, while the constant term gives the y-intercept.

In \(x=3\), x is fixed, so the graph is vertical and parallel to the y-axis; therefore, its slope is undefined. Since it is not \(x=0\), it does not meet the y-axis. Exam tip: \(x=k\) represents a vertical line.

Rewriting option A gives \(y=-2x+4\). Therefore, its slope is \(-2\), which is negative, and its y-intercept is 4. Option D has a negative slope but intercepts the y-axis at 2. Exam tip: convert the equation to \(y=mx+c\) first.

The given line has slope \(-3\) and y-intercept \(7\). For perpendicular non-vertical lines, the product of slopes is \(-1\), so the required slope is \(\frac{1}{3}\). Keeping intercept \(7\) gives option A. Option B is parallel, not perpendicular. Exam tip: in \(y=mx+c\), \(c\) is the y-intercept.

For \(y=mx+c\), a line falls left to right when \(m<0\), and it meets the \(y\)-axis above the origin when \(c>0\). In option B, \(m=-4\) and \(c=7\). Option A has the correct intercept sign but a positive slope. Exam tip: check the signs of slope and intercept separately.

The slope is \(\frac{\Delta y}{\Delta x}=\frac{28}{4}=7\). The \(y\)-intercept is \(-10\), so in slope-intercept form \(y=mx+c\), substituting \(m=7\) and \(c=-10\) gives \(y=7x-10\). In \(y=28x-10\), the slope is 28, while in \(y=-7x-10\), \(y\) decreases as \(x\) increases. Exam tip: find slope by calculating \(\Delta y/\Delta x\).

When (x) increases by 5, (y) decreases by 30, so the slope is \(m=\frac{-30}{5}=-6\). Since the line passes through (0,14), its (y)-intercept is 14. Hence the equation is \(y=-6x+14\). In \(y=-30x+14\), the intercept is correct but the slope is -30, not -6. Exam tip: find the rate of change using \(\frac{\Delta y}{\Delta x}\).

The line is \,\(y=(a+4)x+(a-10)\). At the origin, \,\(x=0\) and \,\(y=0\), so \,\(a-10=0\), giving \,\(a=10\). The slope is the coefficient of \,\(x\): \,\(a+4=10+4=14\). Hence, 14 is correct. A value such as 12 would result from using an incorrect value of \,\(a\). Exam tip: in \,\(y=mx+c\), \,\(m\) is the slope and \,\(c\) is the y-intercept.

In the form y=mx+c, the y-intercept is c. Here, u-2=6, so u=8. Therefore, the slope is m=2u+1=2(8)+1=17. Option 16 would result from incorrectly omitting the +1 term. Exam tip: first use the y-intercept to find the parameter, then substitute it into the slope expression.

To find a y-intercept, put \(x=0\) in the equation of the line. The y-intercept of the first line is \(b\), while for the second line, putting \(x=0\) gives \(y=8\). Since both lines cut the y-axis at the same point, their y-intercepts must be equal: \(b=8\). The numbers \(6\) and \(-4\) are slopes, not y-intercepts. Exam tip: in \(y=mx+c\), \(c\) is the y-intercept directly.