Understanding Linear Equations and Intersection Points

The question asks for the value of \(k\) when the graphs of two linear equations, \(4x - 2y = 10\) and \(4x + ky = 2\), intersect at a specific point \((a, 4)\). The point of intersection is where both equations are true simultaneously for the same \(x\) and \(y\) values. In this case, the intersection point is given as \((a, 4)\), meaning when \(x=a\), \(y=4\), both equations are satisfied.

Step 1: Using the Intersection Point in the First Equation

Since the point \((a, 4)\) lies on the line represented by the equation \(4x - 2y = 10\), we can substitute the coordinates of this point into the equation. Here, \(x=a\) and \(y=4\).

Substituting \(x=a\) and \(y=4\) into \(4x - 2y = 10\):

\(4(a) - 2(4) = 10\)

Now, we solve this equation to find the value of \(a\):

\(4a - 8 = 10\)

Add 8 to both sides of the equation:

\(4a = 10 + 8\)

\(4a = 18\)

Divide both sides by 4:

\(a = \frac{18}{4}\)

Simplify the fraction:

\(a = \frac{9}{2}\)

So, the x-coordinate of the intersection point is \(a = \frac{9}{2}\).

Step 2: Using the Intersection Point in the Second Equation to Find k

The point \((a, 4)\) also lies on the line represented by the equation \(4x + ky = 2\). Now that we know \(a = \frac{9}{2}\), the intersection point is \(\left(\frac{9}{2}, 4\right)\). We substitute \(x = \frac{9}{2}\) and \(y = 4\) into the second equation:

\(4x + ky = 2\)

Substituting \(x = \frac{9}{2}\) and \(y = 4\):

\(4\left(\frac{9}{2}\right) + k(4) = 2\)

Simplify the first term:

\(2 \times 9 + 4k = 2\)

\(18 + 4k = 2\)

Now, we solve for \(k\). Subtract 18 from both sides of the equation:

\(4k = 2 - 18\)

\(4k = -16\)

Divide both sides by 4:

\(k = \frac{-16}{4}\)

\(k = -4\)

Therefore, the value of \(k\) is -4.

Verifying the Solution

The intersection point is \(\left(\frac{9}{2}, 4\right)\) and \(k = -4\). Let's check if this point satisfies both original equations:

• For \(4x - 2y = 10\): \(4\left(\frac{9}{2}\right) - 2(4) = 2 \times 9 - 8 = 18 - 8 = 10\). This is correct.
• For \(4x + ky = 2\): \(4\left(\frac{9}{2}\right) + (-4)(4) = 2 \times 9 - 16 = 18 - 16 = 2\). This is also correct.

Since the point \(\left(\frac{9}{2}, 4\right)\) satisfies both equations when \(k = -4\), our calculated value for \(k\) is correct.

Summary of Finding k

We used the given intersection point \((a, 4)\) to first find the value of \(a\) by substituting it into the first linear equation \(4x - 2y = 10\). Once \(a\) was found to be \(\frac{9}{2}\), we used the full intersection point \(\left(\frac{9}{2}, 4\right)\) and substituted it into the second linear equation \(4x + ky = 2\). This allowed us to solve for the unknown variable \(k\).

Revision Table: Key Concepts for Linear Equations

Additional Information: Systems of Linear Equations

A system of linear equations consists of two or more linear equations involving the same variables. The solution to a system of two linear equations in two variables (like \(x\) and \(y\)) is the point \((x, y)\) that satisfies both equations. Graphically, this solution is the point of intersection of the lines represented by the equations.

There are three possible outcomes for a system of two linear equations:

• One Unique Solution: The lines intersect at exactly one point. This is the case in our question. The system is called consistent and independent.
• No Solution: The lines are parallel and distinct, meaning they never intersect. The system is called inconsistent. This happens when the slopes are the same but the y-intercepts are different.
• Infinitely Many Solutions: The lines are identical (coincident). Every point on the line is an intersection point. The system is called consistent and dependent. This happens when both the slopes and the y-intercepts are the same.

In this problem, knowing the intersection point allowed us to work backward and find an unknown coefficient \(k\) within one of the equations.