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. 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.

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.

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.

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.

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 \(x\).

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.

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.

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.

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.

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.

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.

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.

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.

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 \((0,c)\).

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 \((0,c)\).

In the form y=mx+c, m is the slope. Here, the coefficient of x is 8, so the slope is 8. Therefore, when x increases by 1, y increases by 8. The number 6 is the y-intercept, which gives the value of y when x=0; it is not the rate of increase. Exam tip: In y=mx+c, identify the coefficient of x to find the slope quickly.

The line is in the form y=mx+c, where m is the slope. Here, m=-6. Therefore, when x increases by 1, the change in y is -6, meaning y decreases by 6. The number 13 is the y-intercept; it does not give the change in y for a one-unit change in x. Exam tip: In y=mx+c, the change in y for an increase of 1 in x is exactly m.

In y=7x-11, the term 7x depends on x, whereas -11 has no x. Therefore, the constant term is -11. The number 7 is the coefficient of x, not the constant term. Exam tip: A term with no variable is the constant term.

In y = -9x + 16, the number multiplying x is -9, so the coefficient of x is -9. The number 16 is the constant term, not the coefficient of x. In this form, -9 also represents the slope of the line. Exam tip: the number that directly multiplies a variable is its coefficient.

In the form \(y=mx+c\), \(c\) is the \(y\)-intercept because putting \(x=0\) gives \(y=c\). In \(y=2x-3\), the constant term is \(-3\), so its \(y\)-intercept is \(-3\). In \(y=-3x+2\), \(-3\) is the slope, not the \(y\)-intercept. Exam tip: To find the \(y\)-intercept, put \(x=0\) in the equation.