Question

find a coefficient a for the linear equation ax-y=4 such as the graph of the equation would pass through the point M(3,5). graph this equation

Answer 1

To find the coefficient \(a\) for the linear equation \(ax - y = 4\) such that the graph passes through the point \(M(3,5)\), we can substitute the coordinates of point \(M\) into the equation.

Given the point \(M(3,5)\):
- Substitute \(x = 3\) and \(y = 5\) into the equation:

\[
a(3) - 5 = 4
\]

This simplifies to:

\[
3a - 5 = 4
\]

Next, solve for \(a\):

\[
3a = 4 + 5 \\
3a = 9 \\
a = \frac{9}{3} \\
a = 3
\]

So, the coefficient \(a\) is \(3\). Therefore, the equation of the line is:

\[
3x - y = 4
\]

Now, let's rewrite it in slope-intercept form (which is \(y = mx + b\)):

\[
-y = 4 - 3x \\
y = 3x - 4
\]

### Graphing the Equation

1. **Identify the y-intercept**: When \(x = 0\),
\[
y = 3(0) - 4 = -4
\]
So, the y-intercept is \( (0, -4) \).

2. **Identify another point using the slope**: The slope \(m\) is \(3\), meaning for every 1 unit you move to the right (positive \(x\)), you move up 3 units (positive \(y\)).
- From the y-intercept, (0, -4) moving to the right \(1\) unit:
\[
x = 1 \Rightarrow y = 3(1) - 4 = 3 - 4 = -1
\]
So another point is \( (1, -1) \).

3. **Plot the points**: You can plot the points \( (0, -4) \) and \( (1, -1) \).

4. **Draw the line**: Connect these points to visualize the line represented by the equation \(3x - y = 4\).

### Summary
The coefficient \(a = 3\), and the linear equation is \(3x - y = 4\). When plotted, this equation will indeed pass through the point \(M(3,5)\).