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. Here, \(m=-5\), and \(y=3\) when \(x=0\); therefore, \(c=3\). Hence, the equation is \(y=-5x+3\). Option D has the correct slope but its \(y\)-intercept is \(-3\). Exam tip: the value of \(y\) when \(x=0\) is the \(y\)-intercept.

In the form y = mx + c, the coefficient m of x is the slope. The coefficient of x is 4 in both equations, so the two lines have the same slope and are parallel. Their y-intercepts are 7 and -2 respectively, so option B is incorrect. Exam tip: To find the slope in y = mx + c, identify the coefficient of x.

In the form y=mx+c, c represents the y-intercept. Both equations have the constant term -1, so both lines meet the y-axis at (0,-1). Their slopes are 6 and -3, so the slopes are not equal and the second slope is negative. Exam tip: In y=mx+c, identify m as the slope and c as the y-intercept.

A line is written as \(y=mx+c\), where \(m\) is the slope and \(c\) is the \(y\)-intercept. Here, \(m=0\) and \(c=-4\), so \(y=0x-4=-4\). Option C has a \(y\)-intercept of \(-4\), but its slope is \(4\), not zero. Exam tip: A line with zero slope is always horizontal.

The slope-intercept form of a line is \(y=mx+c\), where \(m\) is the slope and \(c\) is the \(y\)-intercept. A line through the origin \((0,0)\) has \(c=0\). With slope \(9\), its equation is \(y=9x\), so option A is correct. Although \(y=9x+1\) has slope 9, its \(y\)-intercept is 1, so it does not pass through the origin. Exam tip: for a line through the origin, the constant term is always 0.

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.

In y = mx + c, the coefficient m determines the direction of the line and c determines its y-intercept. Since m > 0, increasing x produces an increase in y, so the line is increasing rather than decreasing or horizontal. Since c = 0, the equation becomes y = mx; when x = 0, y = 0, so the line passes through the origin. Therefore option B correctly combines both conclusions. Option A reverses the slope behavior, option C would require m = 0, and option D is false 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.