Rate of change in 8th grade math

Answer. 2

The slope is $2$ because each time x increases by $1$, y increases by $2$, and $2 \div 1 = 2$.

1. 1Look at any two consecutive rows in the table.
2. 2Using the first two data rows, x goes from 1 to 2 and y goes from 3 to 5.
3. 3Slope is change in y divided by change in x.

Common mistake. A common mistake is computing $\Delta x \div \Delta y$ instead of $\Delta y \div \Delta x$, which would give $\frac{1}{2}$ instead of $2$. Always divide the change in y by the change in x.

The idea underneath. Slope is always the change in y divided by the change in x (Δy ÷ Δx). Even when x increases by 1, you are dividing — it just doesn't change the result. When x increases by more than 1, the division step visibly matters.

The correct statement is that Alex used one y-value instead of the change in y for the slope. The initial value is 5 because y = 5 when x = 0, and the slope is 3 because y increases by 3 each time x increases by 1.

1. 1Start with the initial value. In a table, the initial value is the y-value in the row where x = 0.
2. 2When x = 0, y = 5, so the initial value is 5.
3. 3Now find the slope. Because x goes up by 1 each row, the slope is the repeated change in y.
4. 4Alex's note got the initial value right, but 8 is just the y-value when x = 1. It is not the change in y, so Alex used one y-entry instead of the slope.

Common mistake. A common mistake is using the y-value 8 from one row as the slope. The slope from a table is not a single entry; it is the amount the y-values change between rows when x increases by 1.

The idea underneath. When x increases by 1 in a table, the slope is the constant step size in the y-column, while the initial value is the y-entry where x = 0.

The initial value is 3, so $b = 3$. Using the other point would give the same value because both points are on the same line.

1. 1Look at the points and slope. We can use the point $(3, 9)$, so $x = 3$, $y = 9$, and $m = 2$.
2. 2Substitute those values into $b = y - mx$.
3. 3Find $mx$ first.
4. 4Subtract 6 from 9 to find $b$.

Common mistake. A common mistake is entering 6 as $b$. That 6 is only the value of $mx$, so you still need to subtract it from 9 to get $b = 3$.

The idea underneath. Once the slope is known, $b$ is whatever remains of $y$ after removing $mx$, so any point on the line should give the same initial value.

Answer. \frac{1}{2}

The initial value is $\frac{1}{2}$ (also accepted as 0.5). The line through (3, 5) and (5, 8) has slope $\frac{3}{2}$, and substituting (3, 5) into b = y − mx gives $\frac{1}{2}$.

1. 1The two ordered pairs lie on the same line. Start by finding the slope m using the change in y over the change in x.
2. 2Calculate the slope from the two points.
3. 3Substitute the slope and the point (3, 5) into b = y − mx.
4. 4Rewrite 5 as $\frac{10}{2}$ so both terms share a denominator of 2, then subtract to get $b = \frac{1}{2}$, which can also be written as 0.5.

Common mistake. A common mistake is to stop after computing the slope and enter $\frac{3}{2}$ as the answer. The slope is m, not b. You must still substitute a point into b = y − mx, which gives b = $\frac{1}{2}$, not $\frac{3}{2}$.

The idea underneath. The initial value is the y-value when x = 0, so once you know the slope, any point on the line lets you work backward to b.