Find the equation of a straight line parallel to the line 2x - y = 5 and having intercept of 5

1. Analyze the question

We are asked to find the equation of a line that satisfies two conditions:

• It is parallel to the line $2x - y = 5$.
• It has a y-intercept of $5$.

2. Explain the concept

• Parallel Lines: Two lines are parallel if and only if they have the same gradient (slope). To find the gradient, we rewrite the equation in the slope-intercept form: $y = mx + c$, where $m$ is the gradient and $c$ is the y-intercept.
• Y-intercept: The y-intercept is the value of $y$ when $x = 0$. In the form $y = mx + c$, the constant $c$ represents the y-intercept.

3. Show the solution

Step 1: Find the gradient of the given line. The given equation is $2x - y = 5$. Rearrange it to make $y$ the subject of the formula: $-y = -2x + 5$ Multiply through by $-1$: $y = 2x - 5$ Comparing this to $y = mx + c$, the gradient ($m$) is 2.

Step 2: Form the new equation. Since the new line is parallel to the given line, it must also have a gradient ($m$) of 2. We are given that the y-intercept ($c$) is 5. Using the slope-intercept form $y = mx + c$: $y = 2x + 5$

Step 3: Rearrange to match the options. Looking at the options provided, they are written in the form $Ax + By = C$. Let's rearrange $y = 2x + 5$: $-2x + y = 5$ Or, multiplying by $-1$: $2x - y = -5$

4. Identify the answer

Comparing our result ($2x - y = -5$) with the given options: A. $2x + y = 5$ B. $2x + y = -5$ C. $2x - y = -5$ D. $2x - y = 5$

Our calculated equation matches option C.

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5. Tip

When dealing with parallel lines in the form $Ax + By = C$, the coefficients of $x$ and $y$ (the $A$ and $B$ parts) remain the same. Only the constant term ($C$) changes. So, for $2x - y = 5$, any parallel line will look like $2x - y = k$. Since the intercept is 5, we plug in $(0, 5)$ into $2x - y = k$: $2(0) - 5 = k \implies k = -5$. Thus, $2x - y = -5$.