Finding the Intersection Point of Linear Equations

The problem asks us to find the point where two given linear equations intersect and then determine which of the provided options represents an equation whose graph also passes through this intersection point. The intersection point of two graphs is the coordinate pair \((x, y)\) that satisfies both equations simultaneously.

The two given equations are:

1. \(4x + \frac{1}{3}y = \frac{8}{3}\)
2. \(\frac{1}{2}x + \frac{3}{4}y + \frac{5}{2} = 0\)

Simplifying the Linear Equations

To make the equations easier to solve, let's clear the fractions from each equation.

• For the first equation, multiply the entire equation by the least common multiple of the denominators (which is 3):
  \(3 \times (4x + \frac{1}{3}y) = 3 \times \frac{8}{3}\)
  \(12x + y = 8\) (Let's call this Equation 3)
• For the second equation, multiply the entire equation by the least common multiple of the denominators (which is 4):
  \(4 \times (\frac{1}{2}x + \frac{3}{4}y + \frac{5}{2}) = 4 \times 0\)
  \(2x + 3y + 10 = 0\) (Let's call this Equation 4)

Now we have a simpler system of linear equations:

1. \(12x + y = 8\)
2. \(2x + 3y + 10 = 0\)

Solving the System to Find the Intersection Point P

We can use either the substitution method or the elimination method to find the point of intersection P\((x, y)\). Let's use the substitution method.

From Equation 3, we can easily express \(y\) in terms of \(x\):
\(y = 8 - 12x\)

Now substitute this expression for \(y\) into Equation 4:

\(2x + 3(8 - 12x) + 10 = 0\)

Distribute the 3:

\(2x + 24 - 36x + 10 = 0\)

Combine like terms:

\((2x - 36x) + (24 + 10) = 0\)
\(-34x + 34 = 0\)

Isolate the term with \(x\):

\(-34x = -34\)

Solve for \(x\):

\(x = \frac{-34}{-34}\)
\(x = 1\)

Now substitute the value of \(x\) back into the equation \(y = 8 - 12x\) to find the value of \(y\):

\(y = 8 - 12(1)\)
\(y = 8 - 12\)
\(y = -4\)

So, the intersection point P has coordinates \((1, -4)\).

Checking which Equation also Passes Through P

The problem states that point P also lies on the graph of one of the given equations in the options. This means that the coordinates of P, \((1, -4)\), must satisfy that equation. We will substitute \(x=1\) and \(y=-4\) into each option to find which one holds true (equals zero).

The substitution shows that only the second option, \(3x - y - 7 = 0\), is satisfied by the point \((1, -4)\).

Therefore, the point P\((1, -4)\) also lies on the graph of the equation \(3x - y - 7 = 0\).

Revision Table: Key Steps to Find the Intersection Point

Additional Information on Linear Equations and Intersection Points

A linear equation in two variables, like \(Ax + By + C = 0\), represents a straight line when graphed on a coordinate plane. The intersection point of two lines is the single point that lies on both lines simultaneously. It is the unique solution \((x, y)\) that satisfies both equations in the system.

If two lines are parallel and distinct, they have no intersection point, and the system of equations has no solution. If two lines are identical (coincident), they intersect at every point, and the system has infinitely many solutions.

The method used here, solving the system of equations, is a standard algebraic approach to find the intersection point of lines. Another way is to graph both lines and visually identify the point of intersection, but this can be less precise.

Checking if a point lies on a line involves substituting the point's coordinates into the equation of the line. If the equation holds true (both sides are equal), the point is on the line; otherwise, it is not.