Slope and y-intercept

Substituting \(x=0\) in \(y=-x+8\) gives \(y=-(0)+8=8\). Hence, \(8\) is correct; it is also the \(y\)-intercept of the line. \(-1\) is the coefficient of \(x\), not the value of \(y\) when \(x=0\). Exam tip: To find the \(y\)-intercept, always put \(x=0\).

The equation is y=14-7x. Substituting x=0 gives y=14-7(0)=14. Therefore, the value of y at x=0 is 14, which is also the y-intercept of the line. The number -7 is the slope, not the value of y at x=0. Exam tip: To find the y-intercept, always put x=0.

In the line equation y=mx+c, m represents the slope and c represents the y-intercept. Here m<0, so the slope is negative, and c<0, so the y-intercept is also negative. Therefore, option B is correct. In option A, the y-intercept is negative, but the slope is incorrectly stated as positive. Exam tip: In y=mx+c, the sign of m gives the slope’s sign and the sign of c gives the y-intercept’s sign.

The governing concept is the slope-intercept form y = mx + c. The coefficient m tells how y changes when x changes, while c gives the y-intercept. Since m > 0, a positive increase in x produces a positive change in y, so the line rises from left to right. Since c = 0, the equation becomes y = mx. Substituting x = 0 gives y = 0, so the line passes through (0, 0), the origin. Therefore option B correctly states both properties. Option A reverses the effect of the slope, option C would require m = 0, and option D is wrong because the y-intercept is zero, not negative.

Subtracting 4 from both sides gives y=3x-4. To find the y-intercept, put x=0; then y=-4. Thus, the graph cuts the y-axis at -4. The number 3 is the slope, not the y-intercept. Exam tip: Write a linear equation as y=mx+c; the value of c is the y-intercept.

Adding 5 to both sides gives \(y=-2x+5\). In the linear form \(y=mx+c\), the coefficient of \(x\), namely \(m\), is the slope. Therefore, the slope is \(-2\). Here, \(5\) is the y-intercept, not the slope. Exam tip: Rewrite the equation in \(y=mx+c\) form and identify the coefficient of \(x\).

In the form \(y=ax-3\), the coefficient of \(x\), namely \(a\), is the slope. Since the given slope is \(-6\), substituting \(a=-6\) gives \(y=-6x-3\). In option C, \(-6\) is the constant term, not the slope. Exam tip: In \(y=mx+c\), the coefficient of \(x\), \(m\), is always the slope.

In the form \(y=4x+b\), the constant term \(b\) is the \(y\)-intercept. Since the \(y\)-intercept is 11, substituting \(b=11\) gives \(y=4x+11\). Option B has intercept \(-11\), while option D changes the slope to \(-4\). Exam tip: in \(y=mx+c\), identify \(c\) directly as the \(y\)-intercept.

In y = mx + b, the coefficient of x gives the slope. Both equations have coefficient -3, so both slopes are -3. Their y-intercepts are 4 and -8, which are different. Since the slopes are equal but the intercepts differ, the lines are distinct parallel lines. Option A is correct. Option B is false because the constants differ; option C ignores the coefficient, and option D is false because neither constant is zero.

In the form y=mx+c, c is the y-intercept. Both equations have the constant term -6, so both lines meet the y-axis at (0,-6). Their slopes are 2 and -7, so their slopes are not the same; the second line is decreasing. Exam tip: To find the y-intercept, put x=0.

A line parallel to the x-axis is horizontal, so its y-coordinate remains constant for every point on it. With y-intercept 8, its equation is \(y=8\). In contrast, \(x=8\) is vertical. Exam tip: a constant y-value always indicates a horizontal line.

The equation is \(y=0x+6\), so its slope is \(0\). When \(x\) increases by \(10\), the change in \(y\) is \(\Delta y=m\Delta x=0\times10=0\). The number \(6\) is only the y-intercept; it does not represent the change in \(y\) as \(x\) changes. Exam tip: In \(y=mx+c\), if \(m=0\), the line is horizontal and \(y\) remains constant.

The slope of \(y=\frac{2}{3}x+1\) is \(\frac{2}{3}\). Therefore, when \(x\) increases by 9, the increase in \(y\) is \(\frac{2}{3}\times 9=6\). Option 3 would result only if the slope were \(\frac{1}{3}\). Exam tip: for \(y=mx+c\), use \(\Delta y=m\Delta x\).

In a linear equation written as y=mx+c, the coefficient m tells how much y changes when x changes by one unit. Here the slope is -4/5, so y moves downward as x increases. The negative sign indicates decrease, while the magnitude 4/5 gives the size of the change per unit increase in x.

When x increases by 15, the change in y is \\(\Delta y=\left(-\frac{4}{5}\right)(15)=-12\\). Thus y becomes 12 units smaller, so the requested decrease is 12, not -12. The constant 12 is the y-intercept and does not affect this change. Therefore option C is correct.

Since the point (2,6) lies on the line y=mx-4, substitute x=2 and y=6: 6=2m-4. Hence, 2m=10 and m=5. The value 4 could result from using the constant term -4 with an incorrect sign. Exam tip: When a point lies on a line, substitute both its coordinates directly into the equation.

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.

Use the standard slope-intercept form y = mx + c. Comparing y = -7x + 9 with this form gives m = -7 and c = 9. Because the coefficient of x is negative, the slope is negative and the line falls as x increases. Because the constant term is positive, the y-intercept is positive; in fact, the line crosses the y-axis at (0, 9). Therefore option B is correct. Option A incorrectly treats -7 as positive, option C reverses the signs of both quantities, and option D is impossible because neither -7 nor 9 is zero. Reading the x coefficient and constant term separately prevents the usual 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.