The plane 2 x+3 y-4 z=5 intersects the x -axis at (a, 0,0),...

The plane $2 x+3 y-4 z=5$ intersects the $x$ -axis at $(a, 0,0),$ the $y$ -axis at $(0, b, 0),$ and the $z$ -axis at $(0,0, c) .$ The value of $a+b+c$ is
(A) 1
(B) $\frac{35}{12}$
(C) 5
(D) $\frac{65}{12}$
(E) 9

Key Concepts

Plane Equation in Three-Dimensional Space

A plane in three-dimensional space is typically represented by a linear equation in x, y, and z. This equation defines a flat surface where every point (x, y, z) satisfies the relationship specified by the equation. Understanding the plane's equation forms the basis for analyzing its position and orientation in space.

Axis Intercepts

An axis intercept of a plane is the point where the plane crosses one of the coordinate axes. To find an intercept, set the remaining coordinates (that are not along that axis) to zero. This method determines the specific value on the axis where the plane meets it, providing a clear geometric intersection point.

Substitution Method

The substitution method involves replacing known values for variables in an equation to simplify and isolate the unknown variable. In the case of axis intercepts, setting the appropriate coordinates to zero reduces the plane's equation to a single-variable linear equation, which can then be solved to find the intercept.

Solving Linear Equations

Solving linear equations is a fundamental algebraic skill that involves isolating the variable of interest using operations such as addition, subtraction, multiplication, and division. This process is essential when determining the exact values where a plane intersects a coordinate axis, as each intercept is found by solving a simplified linear equation.

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