if 3x+4y=5, then what is the least value of x²+y²

We are given the equation:

\[
3x + 4y = 5
\]

We need to find the least value of \( x^2 + y^2 \).

This is a problem of minimizing the distance from the origin \((0,0)\) to the line \( 3x + 4y = 5 \). The formula for the perpendicular distance \( d \) from a point \( (x_1, y_1) \) to a line \( Ax + By + C = 0 \) is:

\[
d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}
\]

In our case, the equation of the line is \( 3x + 4y - 5 = 0 \), where \( A = 3 \), \( B = 4 \), and \( C = -5 \). The distance from the origin \((0,0)\) to the line is:

\[
d = \frac{|3(0) + 4(0) - 5|}{\sqrt{3^2 + 4^2}} = \frac{| - 5 |}{\sqrt{9 + 16}} = \frac{5}{5} = 1
\]

Thus, the least distance between the origin and the line is 1. Since the square of the distance is equal to \( x^2 + y^2 \), the least value of \( x^2 + y^2 \) is:

\[
\boxed{1}
\]

Would you like further clarification or details? Here are some related questions:

1. How do we derive the formula for the distance from a point to a line?
2. What are other ways to solve this type of minimization problem?
3. Can we apply this method to nonlinear equations?
4. How do we solve the system if we add another equation to this problem?
5. What is the geometric significance of minimizing \( x^2 + y^2 \)?

**Tip:** In geometry problems involving distance, minimizing \( x^2 + y^2 \) often involves finding the shortest distance from a point to a line or curve.

If 3x + 4y = 5, then what is the least value of x² + y²?

Discover how to minimize the value of x² + y² for the linear equation 3x + 4y = 5 using geometric methods. This detailed solution covers the perpendicular distance formula and its application in solving minimization problems. Suitable for students in Grades 10-12.

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