Find the equation of a straight line passing through the point (7, 8) making equal intercepts on both axis.

Equation of a Straight Line with Equal Intercepts

A straight line that makes equal intercepts on both the x-axis and y-axis can be expressed in the form:

$$x+y=a$$

where $a$ is the common intercept on both axes.

Step 1: Understanding the intercept form

The general equation of a line making intercepts $a$ on the x-axis and $b$ on the y-axis is:

$$\frac{x}{a}+\frac{y}{b}=1$$

Since the intercepts are equal ($a=b$), the equation simplifies to:

$$\frac{x}{a}+\frac{y}{a}=1⟹x+y=a$$

Step 2: Substituting the given point

The line passes through the point $(7,8)$. Substituting $x=7$ and $y=8$ into the equation:

$$7+8=a⟹a=15$$

Step 3: Final equation

Thus, the equation of the line is:

$$x+y=15$$

Verification

To ensure correctness, let’s verify the intercepts:

• x-intercept: Set $y=0$, then $x=15$.
• y-intercept: Set $x=0$, then $y=15$.

Both intercepts are equal, confirming the solution.

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Alternative Approach (Slope-Intercept Form)

Alternatively, we can derive the equation using the slope-intercept form $y=mx+c$.

Step 1: Find the slope

Since the intercepts are equal, the line passes through $(a,0)$ and $(0,a)$. The slope $m$ is:

$$m=\frac{a-0}{0-a}=-1$$

Step 2: Use point-slope form

Using the point $(7,8)$ and slope $m=-1$:

$$y-8=-1(x-7)⟹y=-x+7+8⟹x+y=15$$

This confirms the earlier result.

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Final Answer

The equation of the straight line passing through $(7,8)$ and making equal intercepts on both axes is:

$x+y=15$