Slope and y-intercept

The line is in the form \(y=6x+b\), where \(b\) is the \(y\)-intercept. Since the \(y\)-intercept is \(13\), substituting \(b=13\) gives \(y=6x+13\). In option B, the intercept is \(-13\), while in option D the slope becomes \(-6\). Exam tip: in \(y=mx+c\), the constant term \(c\) is the \(y\)-intercept.

The slope-intercept form y = mx + b shows that the coefficient of x is the slope. In both equations the coefficient is -4, so their slopes are equal. Their y-intercepts are 6 and -9, so they are not equal. Equal slopes with different intercepts mean the lines are distinct and parallel. Hence option A is correct. The remaining options confuse slope with intercept, incorrectly claim a zero slope, or assume passage through the origin without checking the constant term.

In the form y=mx+c, c is the y-intercept. Both equations have the constant term -8, so both lines meet the y-axis at (0,-8). Their slopes are 3 and -9, so their slopes are not the same; the second line is decreasing. Exam tip: Put x=0 to find the y-intercept.

In the form y = mx + c, c is the y-intercept. For y = 3x, c = 0, so the line passes through the origin. The line y = 3 has y-intercept 3. Exam tip: put x = 0 to identify the y-intercept quickly.

The equation is \(y=0x+9\), so its slope is \(0\). Therefore, when \(x\) increases by \(12\), the change in \(y\) is \(0\times 12=0\). Here, \(9\) is the y-intercept, not the change in \(y\). Exam tip: For \(y=mx+c\), use \(\Delta y=m\Delta x\).

The slope is \(\frac{3}{4}\), meaning that for every increase of 4 units in \(x\), \(y\) increases by 3 units. Hence, when \(x\) increases by 8, the change in \(y\) is \(\frac{3}{4}\times 8=6\). The constant 2 is the y-intercept, so it does not affect the increase. Exam tip: use \(\Delta y=m\Delta x\) to find the change in \(y\).

The slope of the line is \(-\frac{2}{7}\). This means that for every increase of \(7\) in \(x\), \(y\) decreases by \(2\). Since \(21=3\times7\), an increase of \(21\) in \(x\) makes \(y\) decrease by \(3\times2=6\). Therefore, \(6\) is correct. Exam tip: A negative slope means that \(y\) decreases as \(x\) increases.

Substitute the given point \((3,9)\) into \(y=mx-6\): \(9=3m-6\). Hence, \(3m=15\), so \(m=5\). Therefore, the slope is \(5\). If \(m=4\), the corresponding \(y\)-value would be 6, not 9. Exam tip: To use a point on a line, substitute both its coordinates into the line equation.

Because (5, 11) lies on y = 4x + c, substitute x = 5 and y = 11 into the equation: 11 = 4(5) + c = 20 + c. Subtracting 20 from both sides gives c = 11 - 20 = -9. In y = mx + c, c is the y-intercept, the value of y when x = 0. Therefore the y-intercept is -9, so option B is correct. Option A results from reversing the subtraction, while C and D use an incorrect operation or sign.

The slope-intercept form of a line is \(y=mx+c\), where \(m\) is the slope and \(c\) is the y-intercept. Here, \(m=-3\), and \(y=15\) when \(x=0\), so \(c=15\). Therefore, the line is \(y=-3x+15\). Option A has y-intercept 15, but its slope is \(+3\), not \(-3\). Exam tip: In \(y=mx+c\), the coefficient of \(x\) is the slope.

The slope-intercept form of a line is \(y=mx+c\), where \(m\) is the slope and \(c\) is the y-intercept. Here, \(m=8\) and \(y=-10\) when \(x=0\), so \(c=-10\). Therefore, the line is \(y=8x-10\). Option C has the correct y-intercept but its slope is \(-8\). Exam tip: Substitute \(x=0\); the resulting value of \(y\) is the y-intercept.

A line in slope-intercept form is y = mx + c, where m is the slope and c is the y-intercept. For the two equations, the slopes are 10 and -10, so they are different; their constant terms are both 3, so both meet the y-axis at (0, 3). Therefore, option B is correct. Option A reverses the comparison, while C and D contradict the equations.

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 their slopes are the same. Their y-intercepts are 14 and -4 respectively, so they are different. Lines with equal slopes and different y-intercepts are parallel. Exam tip: identify the coefficient of x for slope and the constant term for the y-intercept.

The governing form is y = mx + c. In y = -9x + 11, the coefficient of x is m = -9, so the slope is negative. The constant term is c = 11, so the y-intercept is positive; specifically, the line meets the y-axis at (0, 11). Therefore option B is the only correct answer. Option A incorrectly changes the sign of the slope, option C reverses both signs, and option D ignores that both displayed values are nonzero. The sign of the x coefficient and the sign of the constant must be read independently rather than inferred from the overall appearance of the equation.

In \(y=mx+c\), \(m\) is the slope and \(c\) is the \(y\)-intercept. Substituting \(x=0\) gives \(y=5\), so A is correct; \(-2\) is the slope. Exam tip: identify the constant term first.

The line is \(y=mx-7\), where \(m\) represents the slope. Since the slope is 5, \(m=5\). Substituting \(x=4\), we get \(y=5\times4-7=20-7=13\). Therefore, option B is correct. \(20\) is only \(5\times4\); the subtraction of 7 is still required. Exam tip: First replace \(m\) with the given slope, then substitute the value of \(x\).

In slope-intercept form \(y=mx+c\), \(c\) is the constant term. Setting \(x=0\) gives \(y=c\), so it is the y-intercept. The slope is \(m\), not \(c\). Exam tip: put \(x=0\) to identify the y-intercept.

The slope of \(y=\frac{5}{4}x-2\) is \(\frac{5}{4}\). This means that for every increase of 4 units in \(x\), \(y\) increases by 5 units. Therefore, when \(x\) increases by 12 units, \(\Delta y=\frac{5}{4}\times 12=15\). The constant term \(-2\) gives the y-intercept and does not affect the change in \(y\). Exam tip: for \(y=mx+c\), use \(\Delta y=m\Delta x\).

In the form y=mx+c, the y-intercept is c. Since the y-intercept is 7, c=7. Substituting x=6 gives y=-2(6)+7=-12+7=-5. Option 5 results from missing the negative sign. Exam tip: first identify c from the y-intercept, then substitute the given value of x.

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.