Slope and y-intercept

In \(y=mx+c\), \(m\) is the slope and \(c\) is the y-intercept. Here \(m=-3\), so the line falls as x increases, while \(c=7\) means it meets the y-axis at \(7\). Exam tip: a negative slope always indicates a downward line.

In \(y=mx+c\), putting \(x=0\) gives \(y=c\), so the y-axis point is \((0,c)\). Therefore, both lines have the same y-intercept only when \(c_1=c_2\). \(m_1=m_2\) indicates equal slopes, usually parallel lines. Exam tip: set \(x=0\) to identify a y-intercept.

Substitute the point (-2,1) into y=mx+7: 1=m(-2)+7. Thus, 1=-2m+7, so -2m=-6 and m=3. Therefore, the slope is 3. If m were -3, the y-coordinate would be 13, not 1. Exam tip: substitute both coordinates carefully, especially when the x-coordinate is negative.

In \(x=4\), the x-coordinate remains fixed, so the graph is a vertical line. Since \(\Delta x=0\), its slope is undefined, and it does not meet the y-axis. Exam tip: \(x=a\) always represents a vertical line.

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.