Slope and y-intercept

Correct answer: A. 4

In the form \(y=mx+c\), the coefficient of \(x\), namely \(m\), is the slope. Here, in \(y=4x+3\), \(m=4\), so the slope is 4. The number 3 is the \(y\)-intercept, not the slope. Exam tip: identify the coefficient of \(x\) to find the slope.

Correct answer: B. -6

In the standard form \(y=mx+c\), \(c\) is the \(y\)-intercept because putting \(x=0\) gives \(y=c\). Here, in \(y=5x-6\), \(c=-6\); therefore, the \(y\)-intercept is \(-6\). The value \(5\) is the slope, not the intercept. Exam tip: To find the \(y\)-intercept, set \(x=0\).

Correct answer: B. -3

The slope-intercept form of a line is \(y=mx+c\), where \(m\) is the slope. In \(y=-3x+8\), the coefficient of \(x\) is \(-3\), so the slope is \(-3\). The number \(8\) is the y-intercept, not the slope. Exam tip: In \(y=mx+c\), the coefficient of \(x\) gives the slope directly.

Correct answer: D. -4

Write the line in slope-intercept form: \(y=-4x+11\). In \(y=mx+c\), the coefficient of \(x\), namely \(m\), is the slope. Therefore, the slope is \(-4\). Here, 11 is the y-intercept, not the slope. Exam tip: In \(y=a+bx\), the coefficient \(b\) of \(x\) is the slope.

Correct answer: A. 9

The y-intercept is the y-value where the line meets the y-axis, so x=0. Substituting x=0 in y=2x+9 gives y=2(0)+9=9. Therefore, 9 is correct. The number 2 is the coefficient of x, or the slope, not the y-intercept. Exam tip: In the form y=mx+c, the constant term c is the y-intercept.

Correct answer: B. -7

The slope-intercept form of a line is y=mx+c, where the coefficient of x, m, is the slope. In y=-7x+1, m=-7, so the slope is -7. The number 1 is the y-intercept, not the slope. Exam tip: In y=mx+c, identify the number multiplying x to find the slope.

Correct answer: D. \(0\)

A line in the form \(y=mx+c\) has slope \(m\), the coefficient of \(x\). In \(y=0x+14\), the coefficient of \(x\) is \(0\), so the slope is \(0\). The number \(14\) is the y-intercept, not the slope. Exam tip: In \(y=mx+c\), identify the coefficient of \(x\) to find the slope.

Correct answer: A. By the coefficient m of x

In the linear equation y=mx+c, m is the coefficient of x and represents the slope of the line. The constant c does not give the slope; it represents the y-intercept. Exam tip: identify the coefficient multiplying x to find m.

Correct answer: B. Constant term (c)

In the linear form \(y=mx+c\), \(c\) is the constant term and represents the y-intercept. On putting \(x=0\), we get \(y=c\), so the graph meets the y-axis at \(c\). In contrast, \(m\) represents the slope, not the y-intercept. Exam tip: set \(x=0\) to find the y-intercept.

Correct answer: C. \(y=5x+2\)

The slope-intercept form of a line is \(y=mx+c\), where \(m\) is the slope and \(c\) is the \(y\)-intercept. Substituting \(m=5\) and \(c=2\) gives \(y=5x+2\). Therefore, option C is correct. Option B has slope 5, but its \(y\)-intercept is \(-2\). Exam tip: in \(y=mx+c\), the coefficient of \(x\) is the slope and the constant term is the \(y\)-intercept.

Correct answer: D. \(y=-2x+4\)

The slope-intercept form of a line is \(y=mx+c\), where \(m\) is the slope and \(c\) is the \(y\)-intercept. Substituting \(m=-2\) and \(c=4\) gives \(y=-2x+4\). Option A has a \(y\)-intercept of 4, but its slope is \(2\), not \(-2\). Exam tip: the coefficient of \(x\) is the slope, while the constant term is the \(y\)-intercept.

Correct answer: A. 0

In the form y = mx + c, the y-intercept is c. Here, in y = 9x + 0, c = 0, so the y-intercept is 0 and the line passes through the origin (0, 0). The number 9 is the slope, not the y-intercept. Exam tip: To find the y-intercept, put x = 0.

Correct answer: B. -10

The slope-intercept form of a line is \(y=mx+c\), where \(m\) is the slope. In \(y=-10x-3\), the coefficient of \(x\) is \(-10\), so the slope is \(-10\). The value \(-3\) is the y-intercept, not the slope. Exam tip: in \(y=mx+c\), the coefficient of \(x\) is always the slope.

Correct answer: C. 15

To find the y-intercept, put x=0. In y=15+6x, this gives y=15. Therefore, the line crosses the y-axis at 15. Here, 6 is the slope, not the y-intercept. Exam tip: in the form y=mx+c, c is always the y-intercept.

Correct answer: D. \(m=-5,\;c=20\)

Compare the equation with slope-intercept form \(y=mx+c\). The given line can be written as \(y=-5x+20\). Therefore, the coefficient of \(x\) gives the slope \(m=-5\), and the constant term gives the \(y\)-intercept \(c=20\). Option B has the correct intercept but the wrong sign for the slope. Exam tip: In \(y=mx+c\), always check the sign of the coefficient of \(x\).

Correct answer: A. Slope 3, y-intercept -8

The standard form of a line is \(y=mx+c\), where \(m\) is the slope and \(c\) is the y-intercept. In \(y=3x-8\), \(m=3\) and \(c=-8\); therefore, the slope is 3 and the y-intercept is -8. Option B incorrectly interchanges the slope and the intercept. Exam tip: identify the coefficient of \(x\) as the slope and the constant term as the y-intercept.

Correct answer: B. Slope -1, y-intercept 12

The governing concept is the slope-intercept form of a straight line, y = mx + c, where m is the slope and c is the y-intercept. Comparing y = -x + 12 with this standard form, the coefficient of x is -1, so m = -1. The constant term is 12, so the line crosses the y-axis at (0, 12), and its y-intercept is 12. Therefore, option B gives both values correctly. Option A reverses the two quantities, option C changes the sign of the slope, and option D incorrectly treats the numbers as reciprocals or rearranged coefficients. The negative slope also means that y decreases by 1 whenever x increases by 1.

Correct answer: C. \(\frac{2}{3}\)

The slope-intercept form of a line is \(y=mx+c\), where \(m\) is the slope. In \(y=\frac{2}{3}x+5\), the coefficient of \(x\) is \(\frac{2}{3}\), so the slope is \(\frac{2}{3}\). The number 5 is the y-intercept, not the slope. Exam tip: In \(y=mx+c\), the coefficient of \(x\) is always the slope.

Correct answer: B. \(-2\)

A line in the form \(y=mx+c\) has y-intercept \(c\). Here, \(c=-2\), so the y-intercept is \(-2\); the line meets the y-axis at \((0,-2)\). \(-\frac{5}{6}\) is the slope, not the y-intercept. Exam tip: put \(x=0\) to find the y-intercept.

Correct answer: A. Goes upward from left to right

A positive slope means that as the value of x increases, the value of y also increases. Therefore, the line rises from left to right. A line that falls from left to right has a negative slope. Exam tip: If a graph rises as you move to the right, its slope is positive.

Correct answer: B. Goes downward from left to right

A negative slope means that as the value of x increases, the value of y decreases. Therefore, the line moves downward from left to right on the graph. Option A represents a positive slope, where y increases as x increases. Exam tip: if a line falls while moving to the right, its slope is negative.

Correct answer: C. \(y=7\)

A line with slope 0 is horizontal and has the form \(y=\text{constant}\). In \(y=7\), the value of \(y\) remains 7 for every value of \(x\), so its slope is 0. The slopes of \(y=2x+1\) and \(y=-3x+5\) are 2 and −3 respectively. Exam tip: a line of the form \(y=\text{constant}\) always has zero slope.

Correct answer: D. \(y=6x\)

A line of the form \(y=mx+c\) passes through the origin \((0,0)\) only when \(c=0\). In \(y=6x\), the constant term is 0, and substituting \(x=0\) gives \(y=0\). In contrast, \(y=5x+1\) has y-intercept 1, so it does not pass through the origin. Exam tip: To test for the origin, substitute \(x=0\) in the equation.

Correct answer: A. \((0,10)\)

At any point on the y-axis, the x-coordinate is 0. Substituting \(x=0\) in \(y=4x+10\) gives \(y=10\). Hence, the line cuts the y-axis at \((0,10)\). In \((0,4)\), 4 is the slope, not the y-intercept. Exam tip: for \(y=mx+c\), the y-intercept is always \((0,c)\).

Correct answer: B. \((0,7)\)

Every point on the y-axis has x-coordinate 0. Substituting \(x=0\) in the equation gives \(y=-5(0)+7=7\). Therefore, the line cuts the y-axis at \((0,7)\). The value -5 is the slope, not the y-intercept, so \((0,-5)\) is incorrect. Exam tip: In \(y=mx+c\), the y-intercept is always \((0,c)\).