Slope and y-intercept

The origin is \((0,0)\). Substituting \(x=0\) in \(y=mx+c\) gives \(y=c\); for the line to pass through the origin, \(y\) must be 0, so \(c=0\). \(m=0\) only represents a horizontal line. Exam tip: remember that \(c\) 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. Here, \(m=-4\), and \(y=12\) when \(x=0\), so \(c=12\). Therefore, the equation is \(y=-4x+12\). Option A has the correct y-intercept but its slope is \(+4\). Exam tip: Substitute \(x=0\) to identify the y-intercept directly from an equation.

The slope-intercept form of a line is \(y=mx+c\), where \(m\) is the slope and \(c\) is the y-intercept. Here, \(m=6\), and when \(x=0\), \(y=-8\); therefore \(c=-8\). Hence, the equation is \(y=6x-8\). In \(y=-8x+6\), the slope is \(-8\), so it is not correct. Exam tip: in \(y=mx+c\), the coefficient of \(x\) is always the slope.

In the form (y=mx+c), (m) is the slope and (c) is the (y)-intercept. The first line has slope (9), while the second has slope (-9), so their slopes are different. Both equations have constant term (2), so both have (y)-intercept (2). Option D is incorrect because a line passes through the origin only when its (y)-intercept is (0). Exam tip: In (y=mx+c), the coefficient of (x) gives the slope and the constant term gives the (y)-intercept.

In the form \(y=mx+c\), \(m\) is the slope and \(c\) is the y-intercept. Both equations have \(-2\) as the coefficient of \(x\), so their slopes are the same. However, their y-intercepts are \(10\) and \(-3\), respectively, so they are different. Therefore, the lines are parallel, not the same line. Exam tip: in \(y=mx+c\), identify the coefficient of \(x\) for slope and the constant term for the y-intercept.

Compare y = 11x - 4 with y = mx + b. The coefficient of x is m = 11, which is positive, so the slope is positive. The constant term is b = -4, which is negative, so the y-intercept is negative. Therefore option A is correct. The signs must be read separately; the positive coefficient does not make the constant positive, and the negative constant does not change the slope.

Use the slope-intercept form y = mx + c. Comparing y = -7x + 9 with that form gives m = -7 and c = 9. The negative coefficient of x means the slope is negative: the line falls as x increases. The positive constant term means the y-intercept is positive, specifically the line crosses the y-axis at (0, 9). Thus option B is correct. Option A ignores the minus sign on the slope, option C reverses both signs, and option D incorrectly treats nonzero values as zero. Reading the coefficient and constant separately avoids the common sign error.

The y-intercept is 2, so the constant in the equation is \(c=2\). Substituting \(x=4\), we get \(y=3\times4+2=14\). Therefore, 14 is correct. The value 12 is only \(3\times4\); the y-intercept 2 must also be added. Exam tip: in \(y=mx+c\), the y-intercept is always \(c\).

In the line equation \(y=mx-5\), \(m\) represents the slope. Since the slope is 4, \(m=4\). Substituting \(x=3\) gives \(y=4\times3-5=12-5=7\). Therefore, 7 is correct. Getting 9 would result from incorrectly adding the constant term \(-5\). Exam tip: first identify the value of \(m\), then substitute the given value of \(x\).

In the form \(y=mx+c\), \(m\) is the slope and \(c\) is the y-intercept. In option A, \(m=-3\) is negative and \(c=4\). Option C has a negative slope but its y-intercept is \(-4\). Exam tip: check the coefficient of x and the constant term separately.

Dividing both sides of 3y=15x+18 by 3 gives y=5x+6. In the standard form y=mx+c, the coefficient of x is m, which represents the slope. Therefore, the slope is 5. Although 15 is the coefficient of x in the original equation, it must be divided by 3 because the coefficient of y is 3. Exam tip: First rewrite the equation in y=mx+c form, then identify the coefficient of x.

Dividing 5y = 20x - 35 by 5 gives y = 4x - 7. In the form y = mx + c, the constant c is the y-intercept, so the y-intercept is -7. Here, 4 is the slope, not the y-intercept. Exam tip: You can also find the y-intercept by putting x = 0.

Write the equation in slope-intercept form: \(4x+y=13\Rightarrow y=-4x+13\). In the form \(y=mx+c\), the coefficient of \(x\), namely \(m\), is the slope. Therefore, the slope is \(-4\). Here, 13 is the y-intercept, not the slope. Exam tip: Isolate \(y\) first to identify the slope quickly.

To find the y-intercept, set \(x=0\). Substituting \(x=0\) in the equation gives \(-y=18\), so \(y=-18\). Hence, the y-intercept is \(-18\). The close distractor \(18\) results from overlooking the negative sign in \(-y=18\). Exam tip: always put \(x=0\) to find the y-intercept.

Putting \(m=0\) gives \(y=c\). Since the value of \(y\) remains constant, the line is parallel to the \(x\)-axis and meets the \(y\)-axis at \(c\). A line parallel to the \(y\)-axis has the form \(x=a\), not \(y=c\). Exam tip: \(m\) is slope and \(c\) is the y-intercept.

At every point on the \(y\)-axis, \(x=0\). Substituting \(x=0\) in \(y=mx+c\) gives \(y=c\), so \(c\) is the y-intercept. The term \(m\) represents the slope. Exam tip: set \(x=0\) to identify 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 \(m=9\) and \(c=-4\) gives \(y=9x-4\). In option B, the slope and the \(y\)-intercept have been interchanged. Exam tip: in \(y=mx+c\), the coefficient of \(x\) is always the slope.

The slope-intercept form of a line is \(y=mx+c\), where \(m\) is the slope and \(c\) is the \(y\)-intercept. Substituting \(m=-10\) and \(c=8\) gives \(y=-10x+8\). Option A has the wrong sign for the slope because its slope is \(10\). Exam tip: the coefficient of \(x\) gives the slope, while the constant term gives the \(y\)-intercept.

The slope-intercept form of a line is \(y=mx+c\), where \(m\) is the coefficient that represents the slope. Since the slope of \(y=mx-9\) is 6, \(m=6\). The value \(-9\) is the y-intercept, not the slope. Exam tip: In \(y=mx+c\), the coefficient of \(x\) is always the slope.

To find the \(y\)-intercept of \(y=-5x+c\), put \(x=0\). Then \(y=-5(0)+c=c\). Since the given point is \((0,12)\), \(c=12\). The value \(-5\) is the slope of the line, not its \(y\)-intercept. Exam tip: in \(y=mx+c\), \(c\) is the \(y\)-intercept directly.