A line has a slope of 5/7 and includes the points (-7, 5) and (0, j). What is the value of j?
Understand the Problem
The question is asking for the value of 'j' for the given line, which has a specified slope and passes through two points, one of which includes 'j'. We will use the formula for the slope of a line to solve for 'j'.
Answer
$j = 10$
Answer for screen readers
$j = 10$
Steps to Solve
- Identify the given information
We have two points: $(-7, 5)$ and $(0, j)$. The slope $m$ is given as $\frac{5}{7}$.
- Use the slope formula
The formula for the slope between two points $(x_1, y_1)$ and $(x_2, y_2)$ is: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$
In our case:
- $(x_1, y_1) = (-7, 5)$
- $(x_2, y_2) = (0, j)$
- Substitute the values into the slope formula
Substituting the known values, we set up the equation: $$ \frac{j - 5}{0 - (-7)} = \frac{5}{7} $$
Simplifying the denominator, we have: $$ \frac{j - 5}{7} = \frac{5}{7} $$
- Cross-multiply to solve for 'j'
Cross-multiplying gives: $$ 7(j - 5) = 5 \cdot 7 $$ $$ 7(j - 5) = 35 $$
- Distribute and simplify
Distributing $7$: $$ 7j - 35 = 35 $$
Next, we add $35$ to both sides: $$ 7j = 35 + 35 $$ $$ 7j = 70 $$
- Solve for 'j'
Finally, divide both sides by $7$: $$ j = \frac{70}{7} $$ $$ j = 10 $$
$j = 10$
More Information
The value of $j$ is determined by using the slope formula and relates to the concept of linear relationships in coordinate geometry. In this case, the slope describes how steep the line is between two points.
Tips
- Forgetting to simplify the fraction when using the slope formula.
- Not properly substituting the values of the points into the slope formula.
- Neglecting to check the signs while solving the equation, especially during distribution.
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