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. Start with \\(6x+3y=18\\). To isolate \\(y\\), subtract \\(6x\\) from both sides, obtaining \\(3y=-6x+18\\). Dividing every term by 3 gives \\(y=-2x+6\\).

Comparing this result with \\(y=mx+c\\), the coefficient of \\(x\\) is \\(m=-2\\), and the constant term is \\(c=6\\). The line therefore crosses the y-axis at \\(6\\), when \\(x=0\\), and falls by 2 units in \\(y\\) for every unit increase in \\(x\\). Thus option A is correct. A positive slope would ignore the negative sign produced when \\(6x\\) is moved to the other side.

A line in the form y = mx + c has slope m, the coefficient of x. Here, the coefficient of x is k - 3, so k - 3 = 5. Adding 3 to both sides gives k = 8. Option 2 results from moving 3 to the wrong side. Exam tip: In y = mx + c, identify the complete coefficient of x to find the slope.

The equation is \(y=4x+c\), where \(c\) represents the \(y\)-intercept. Substituting the given point \((3,19)\) gives \(19=4\times3+c\). Thus, \(19=12+c\), so \(c=7\). Hence, the \(y\)-intercept is 7. Option 19 is the y-coordinate of the given point, not the intercept. Exam tip: In \(y=mx+c\), \(c\) directly represents the \(y\)-intercept.

In slope-intercept form \(y=mx+c\), \(m\) is the slope and \(c\) is the y-intercept. For \(y=-3x+5\), \(m=-3\) is negative and \(c=5\) is positive. Although \(y=-3x-5\) has a negative slope, its y-intercept is negative. Exam tip: check the coefficient of \(x\) and the constant separately.

Rewriting \(4x+2y=8\) gives \(y=-2x+4\), so the slope is \(-2\) and the y-intercept is \(4\). Option C has a positive intercept but also a positive slope. In exams, compare equations using \(y=mx+c\).

The correct equation is \(y=-2x+3\). Its slope is \(-2\), so the line falls from left to right, and its y-intercept is 3. \(y=-2x-3\) has a negative intercept. Exam tip: in \(y=mx+c\), check the signs of \(m\) and \(c\).

The answer is option C, 3. The slope is the coefficient of x after the equation is written in the form y=mx+c. Starting with ay=12x-18, divide every term by a: y=(12/a)x-18/a. Therefore the slope m is 12/a. The question says the slope is 4, so 12/a=4. Multiplying by a gives 12=4a, and dividing by 4 gives a=3. Option A, 2, would give slope 12/2=6, so it is wrong. Option B, 4, would give slope 12/4=3, so it is also wrong. Option C, 3, gives slope 12/3=4, exactly as required. Option D, 6, would give slope 12/6=2. Thus C is correct. Remember: first isolate y; then the coefficient of x is the slope.

The direct answer is option D: for every value of \(p\), the y-intercept is 5. The y-intercept is the value of \(y\) when \(x=0\). Starting with \(5y=px+25\), divide every term by 5: \(y=\frac{p}{5}x+5\). This is the form \(y=mx+c\), where \(m\) is the slope and \(c\) is the y-intercept. Thus the slope is \(\frac p5\), which changes with \(p\), but the constant term remains 5. Option A, only \(p=5\), is wrong because no special value of \(p\) is needed. Option B, only \(p=25\), is similarly unsupported. Option C, no \(p\), is wrong because every allowed value of \(p\) leaves the constant term unchanged. Option D is correct because putting \(x=0\) directly gives \(5y=25\), hence \(y=5\), regardless of \(p\). Exam cue: in a line written as \(y=mx+c\), read the y-intercept from the constant term.

In the form \(y=mx+c\), \(m\) is the slope. Here, the coefficient of \(x\) is \(2a-1\), so \(2a-1=7\). Hence \(2a=8\) and \(a=4\). If \(a=3\), the slope would be \(2(3)-1=5\), not 7. Exam tip: In \(y=mx+c\), identify the coefficient of \(x\) as the slope.

To identify the slope and y-intercept, write the equation in the form y=mx+c. Dividing every term of 4y=8x-28 by 4 gives y=2x-7. Hence, m=2 is the slope and c=-7 is the y-intercept. Option A incorrectly uses the coefficients before dividing by 4. Exam tip: divide every term by the coefficient of y before identifying m and c.

Rewrite \(ax+by=c\) as \(y=-\frac{a}{b}x+\frac{c}{b}\). In \(y=mx+k\), the coefficient of \(x\) is the slope, so it is \(-\frac{a}{b}\). \(\frac{c}{b}\) is the y-intercept, not the slope. Exam tip: compare with slope-intercept form.

In \(x=4\), the x-coordinate remains fixed for every point, so the graph is vertical and parallel to the y-axis. \(y=4\) is horizontal instead. Exam tip: an equation with a fixed x-value always represents a vertical line.

In the slope-intercept form \(y=mx+c\), the coefficient of \(x\) is the slope \(m\). Here, the slope is \(3k+2\). Since \(3k+2=11\), we get \(3k=9\) and hence \(k=3\). Therefore, option B is correct. If \(k=4\), the slope would be \(14\), not \(11\). Exam tip: In \(y=mx+c\), identify \(m\) directly as the slope.

In \(y=mx+c\), \(m\) is the slope and \(c\) is the \(y\)-intercept. A parallel line needs slope \(3\), and the same intercept requires \(c=-5\). Thus only \(y=3x-5\) fits. Exam tip: check slope and constant separately.

The point \((-2,3)\) lies on the line \(y=mx+9\). Substituting \(x=-2\) and \(y=3\) gives \(3=-2m+9\). Hence, \(-2m=-6\), so \(m=3\). If \(-3\) is used, the point gives \(y=15\), so it is incorrect. Exam tip: To find an unknown slope, substitute the coordinates of the given point into the line equation.

A line has the form \(y=mx+c\), where \(m\) is the slope and \(c\) is the y-intercept. In A, both lines have \(c=5\), but their slopes are \(2\) and \(-3\). Exam tip: compare constant terms first.

The slope is \(\frac{3}{4}\), so for every increase of 4 units in \(x\), \(y\) increases by 3 units. Here, \(\Delta x=16-4=12\). Therefore, \(\Delta y=\frac{3}{4}\times12=9\). Hence, \(y\) increases by 9. Note that 12 is the change in \(x\), not in \(y\). Exam tip: use \(\Delta y=m\Delta x\) for total change; the constant term \(-8\) does not affect the change.

The slope is \(-\frac{5}{6}\), so for every increase of 6 units in \(x\), \(y\) decreases by 5 units. Here, \(\Delta x=18\), so \(\Delta y=-\frac{5}{6}\times18=-15\). Therefore, \(y\) decreases by \(15\) units in total. The value \(20\) is the y-intercept, not the change in \(y\). Exam tip: find total change by multiplying the slope by \(\Delta x\).

Solve the given equation for y: 9x+3y−27=0, so 3y=−9x+27 and y=−3x+9. In the form y=mx+c, m is the slope and c is the y-intercept. Therefore, the slope is −3 and the y-intercept is 9. Option B has the correct intercept but the wrong sign for the slope. Exam tip: Rewrite the equation in y=mx+c form before identifying m and c.

In \(y=mx+c\), \(m\) is the slope. Parallel lines have equal slopes; the given line has slope \(-3\). Option A also has slope \(-3\) but a different intercept, so it is distinct. Option D is the same line. Exam tip: compare the coefficient of \(x\) first.

In the form \(y=mx+c\), \(m\) represents the slope. The slopes of the first and second lines are \(p+1\) and \(6\), respectively. For equal slopes, \(p+1=6\), so \(p=5\). If 6 were chosen, the first slope would be 7, not 6. Exam tip: identify the coefficient of \(x\) when comparing slopes.

In the form \(y=mx+c\), the constant term \(c\) is the \(y\)-intercept. The \(y\)-intercept of \(y=-4x+q\) is \(q\), while that of \(y=7x-2\) is \(-2\). Since the intercepts are equal, \(q=-2\). Note that \(-4\) is the slope of the first line, not its \(y\)-intercept. Exam tip: put \(x=0\) to find a \(y\)-intercept.

A line has the form y=mx+c, where c is its y-intercept. For the line to pass through the origin (0,0), its y-intercept must be 0. Hence, r+1=0, so r=-1. For example, if r=1, the y-intercept is 2, so that option is incorrect. Exam tip: In questions involving the origin, directly substitute x=0 and y=0.

Write the equation in slope-intercept form. In (y=(a-2)x+a), the slope is (a-2) and the y-intercept is a. If they were equal, then (a-2=a). Subtracting a from both sides gives (-2=0), which is impossible. Hence, no value of a satisfies the condition. Even for 0, 1, or 2, the slope remains 2 less than the y-intercept. Exam tip: in y=mx+c, the coefficient of x is the slope and the constant term is the y-intercept.

Substitute the point \((1,5)\) into \(y=2x+b\): \(5=2(1)+b\). Hence, \(b=5-2=3\). To verify, when \(x=3\), \(y=2(3)+3=9\), so the second point \((3,9)\) is also satisfied. Here, \(2\) is the slope, whereas \(b\) is the y-intercept. Exam tip: Substitute the coordinates of a given point into the line equation to find an unknown constant.