Slope and y-intercept

Dividing both sides of 2y=10x+14 by 2 gives y=5x+7. In y=mx+c, the coefficient of x is m, which represents the slope; hence the slope is 5. Here, 7 is the y-intercept, not the slope. Exam tip: First simplify the equation into y=mx+c form, then identify the coefficient of x.

\(y=5\) can be written as \(y=0x+5\). Hence its slope is 0 and its y-intercept is 5, so it is parallel to the x-axis. \(x=5\) is a vertical line. Exam tip: any equation \(y=k\) represents a horizontal line.

Write the equation in slope-intercept form: \(3x+y=11\Rightarrow y=-3x+11\). In the form \(y=mx+c\), the coefficient of \(x\), namely \(m\), is the slope. Hence, the slope is \(-3\). Choosing \(3\) would miss the sign change when \(3x\) is moved to the other side. Exam tip: First rearrange a linear equation into \(y=mx+c\) before identifying its slope.

To find the y-intercept, put x=0. In 5x-y=15, this gives -y=15, so y=-15. Hence, the y-intercept is -15. Option 15 can result from missing the sign change while solving for y. Exam tip: For the y-intercept of a linear equation, always substitute x=0.

Slope-intercept form is \(y=mx+c\), where \(m\) is the slope and \(c\) is the y-intercept. In option A, y is already isolated. Option B must first be rearranged to \(y=-3x+7\). Exam tip: check whether y stands alone.

At every point on the \(y\)-axis, the \(x\)-coordinate is \(0\). Substituting \(x=0\) in \(y=-4x+13\) gives \(y=13\), so the line meets the \(y\)-axis at \((0,13)\). The point \((13,0)\) lies on the \(x\)-axis, so it cannot be the \(y\)-intercept. Exam tip: In \(y=mx+c\), the \(y\)-intercept is \((0,c)\).

The slope-intercept form of a line is \(y=mx+c\), where \(m\) is the slope and \(c\) is the \(y\)-intercept. Substituting \(m=7\) and \(c=-2\) gives \(y=7x-2\). In option C, the intercept has the wrong sign because its \(y\)-intercept is \(+2\). 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 slope and \(c\) is the \(y\)-intercept. Substituting \(m=-8\) and \(c=5\) gives \(y=-8x+5\). Option A has the correct \(y\)-intercept, but its slope is \(8\), not \(-8\). Exam tip: the coefficient of \(x\) is the slope, while the constant term is the \(y\)-intercept.

A line in the form y=mx+c has m as the coefficient of x, and this coefficient is its slope. Since the slope is given as 4, m=4. The constant term -6 represents the y-intercept, not the value of m. Exam tip: In y=mx+c, identify the slope by looking at the coefficient of x.

For the line \(y=-3x+c\), the \(y\)-axis has \(x=0\). Hence \(y=-3(0)+c=c\). Since the line meets the \(y\)-axis at \((0,9)\), its \(y\)-coordinate is 9, so \(c=9\). Here, \(-3\) is the slope, not the \(y\)-intercept. Exam tip: in \(y=mx+c\), \(c\) directly gives the \(y\)-intercept.

In \(y=mx+c\), \(m\) represents the slope. Parallel lines have equal slopes. The given line has slope 2, and option A also has slope 2. Option B has slope −2, so it is not parallel. Exam tip: changing only the constant term gives a parallel line.

In \(y=mx+c\), \(c\) is the y-intercept and \(m\) is the slope. For \(y=-2x+5\), \(c=5\) and \(m=-2\), so the slope is negative. Option B has a positive slope. Exam tip: put \(x=0\) to check the y-intercept.

For the \(y\)-intercept, put \(h=0\): \(T=18-2(0)=18\). Thus, the initial temperature is 18°C. Option B misreads the slope, which is \(-2\). Exam tip: the constant term gives the \(y\)-intercept.

To find the y-intercept, put x=0. Then 2(0)+3y=12, so 3y=12 and y=4. Hence, the line meets the y-axis at 4, so the correct answer is 4. The number 12 is only the constant in the equation, not the y-intercept. Exam tip: always set x=0 to find the y-intercept.

Write the equation in slope-intercept form: \(4x+2y=10\Rightarrow 2y=-4x+10\Rightarrow y=-2x+5\). In the form \(y=mx+c\), the coefficient of \(x\) is the slope \(m\). Hence, the slope is \(-2\). Here, \(5\) is the y-intercept, not the slope. Exam tip: Rearrange a linear equation into \(y=mx+c\) before identifying its slope.

The slope of \(y=\frac{5}{2}x-7\) is \(\frac{5}{2}\). Therefore, when \(x\) increases by \(2\), the change in \(y\) is \(\frac{5}{2}\times2=5\). Hence, \(5\) is correct. The \(-7\) is the y-intercept and does not affect the rate of increase. Exam tip: for \(y=mx+c\), use \(\Delta y=m\Delta x\).

The slope is \(-\frac{3}{4}\), meaning that for every increase of 4 units in \(x\), \(y\) decreases by 3 units. An increase of 8 in \(x\) is two such increases, so \(y\) decreases by \(2\times3=6\). Option 3 would apply only when \(x\) increases by 4. Exam tip: use \(\Delta y=m\Delta x\); a negative change in \(y\) indicates a decrease.

Substitute the coordinates of the given point into the line: y=17 and x=3. Thus, 17=3m+2, so 3m=15 and m=5. If m=4, then y=4×3+2=14, which does not match the given y-coordinate 17. Exam tip: To check whether a point lies on a line, substitute its x- and y-coordinates into the equation.

Substitute the point (4,1) into y=-2x+c: 1=-2(4)+c. Thus, 1=-8+c, so c=9. If c were 8, the right-hand side would be 0, not the given y-coordinate 1. Exam tip: To find an unknown constant in a line, substitute the coordinates of the given point into its equation.

The slope-intercept form of a line is \(y=mx+c\), where \(m\) is the slope and \(c\) is the y-intercept. Here, \(m=2\) and \(c=-5\), so the equation is \(y=2x-5\). The option \(y=2x+5\) has the correct slope but the wrong y-intercept, \(+5\). Exam tip: use the point \((0,c)\) to check the sign of the y-intercept directly.