In the xy-plane, the point (p, r) lies on the line with equation y = x + b, where b is a constant. The point with coordinates (2p, 5r) lies on the line with equation y = 2x + b. If p `ne` 0, what is the value of `r/p` ?

Options

A. `2/5`
B. `3/4`
C. `4/3`
D. `5/2`

Explanation

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Use the first line equation
The point \((p, r)\) lies on the line given by the equation \(y = x + b\). Therefore, we can substitute \(x = p\) and \(y = r\) into this equation:

\[ r = p + b \]

### Step 2: Rearrange to find \(b\)
From the equation \(r = p + b\), we can rearrange it to express \(b\) in terms of \(r\) and \(p\):

\[ b = r - p \]

### Step 3: Use the second line equation
The point \((2p, 5r)\) lies on the line given by the equation \(y = 2x + b\). Substituting \(x = 2p\) and \(y = 5r\) into this equation gives us:

\[ 5r = 2(2p) + b \]

### Step 4: Substitute \(b\) in the second equation
Now, we can substitute the expression for \(b\) we found in Step 2 into this equation:

\[ 5r = 4p + (r - p) \]

### Step 5: Simplify the equation
Now, simplify the equation:

\[ 5r = 4p + r - p \]
\[ 5r = 3p + r \]

### Step 6: Rearrange to isolate \(r\)
Next, we can isolate \(r\) by moving \(r\) to the left side:

\[ 5r - r = 3p \]
\[ 4r = 3p \]

### Step 7: Solve for \(\frac{r}{p}\)
Now, divide both sides by \(p\) (since \(p \neq 0\)) to find \(\frac{r}{p}\):

\[ \frac{r}{p} = \frac{3}{4} \]

### Final Answer
Thus, the value of \(\frac{r}{p}\) is:

\[ \frac{r}{p} = \frac{3}{4} \]

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