Find the equation of a line passing through the intersection of the lines `3x+2y=5` and `2x-y=1` and cuts equal intercepts on the axes.

Options

A. `x+y=2`
B. `x+y=-2`
C. `x-y=2`
D. `-x+y=2`

Explanation

To find the equation of a line passing through the intersection of the lines \(3x + 2y = 5\) and \(2x - y = 1\) that cuts equal intercepts on the axes, we can follow these steps:

### Step 1: Find the intersection point of the two lines.

We have the equations:
1. \(3x + 2y = 5\) (Equation 1)
2. \(2x - y = 1\) (Equation 2)

To solve these equations simultaneously, we can use the elimination method. First, we can manipulate Equation 2 to express \(y\) in terms of \(x\).

From Equation 2:
\[ y = 2x - 1 \]

Now, substitute \(y\) in Equation 1:
\[ 3x + 2(2x - 1) = 5 \]
\[ 3x + 4x - 2 = 5 \]
\[ 7x - 2 = 5 \]
\[ 7x = 7 \]
\[ x = 1 \]

Now, substitute \(x = 1\) back into the expression for \(y\):
\[ y = 2(1) - 1 = 1 \]

Thus, the intersection point is \((1, 1)\).

### Step 2: Determine the equation of the line that cuts equal intercepts.

Since the line cuts equal intercepts on the axes, we can express the equation of the line in intercept form:
\[ \frac{x}{a} + \frac{y}{b} = 1 \]
where \(a = b\). Let \(a = b = k\), then the equation becomes:
\[ \frac{x}{k} + \frac{y}{k} = 1 \]
or simplifying, we have:
\[ x + y = k \]

### Step 3: Use the intersection point to find the value of \(k\).

Since the line passes through the point \((1, 1)\), we substitute these values into the equation:
\[ 1 + 1 = k \]
\[ k = 2 \]

### Step 4: Write the final equation of the line.

Substituting \(k\) back into the equation gives:
\[ x + y = 2 \]

This can also be written in standard form:
\[ x + y - 2 = 0 \]

### Final Answer:
The equation of the line is:
\[ x + y - 2 = 0 \]

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