Finding the intersection of a vector line and a linear equation

Key takeaway: This is a coordinate geometry problem in which one line is given in vector form and the other in Cartesian form. The key is to substitute the parametric coordinates into the linear equation, solve for the parameter, and then recover the intersection point.

Gegeven vorm van de lijnen

We have line $k$ in vectorvorm:

$\begin{pmatrix}x\\ y\end{pmatrix}=\begin{pmatrix}-2\\-3\end{pmatrix}+t\begin{pmatrix}-3\\-5\end{pmatrix}$

and line $l$ in cartesische vorm:

$5x-3y=-1.$

To find the snijpunt, we express $x$ and $y$ from the vector equation and substitute them into the equation of line $l$.

From line $k$:

$x=-2-3t, \qquad y=-3-5t.$

Substitutie in de vergelijking

Substitute these expressions into $5x-3y=-1$:

$5(-2-3t)-3(-3-5t)=-1.$

Now expand:

$-10-15t+9+15t=-1.$

The $t$-terms cancel, leaving

$-1=-1.$

This means the two line equations are dependent in the sense that every point on line $k$ satisfies line $l$. So the parameter $t$ is not determined by the substitution step alone.

Wat betekent dit geometrisch?

Because the substitution produces an identity, the line $k$ lies on line $l$ or the given working contains a contradiction in the stated solution path. In the provided worked solution, the parameter is written as $t=-2$, and then the point is calculated as

$x=-2-2\cdot(-3)=4, \qquad y=-3-2\cdot(-5)=7.$

So the stated intersection point is

$S(4,7).$

Controle van de uitwerking

A careful check should always confirm that the point satisfies both equations. Plugging $S(4,7)$ into $5x-3y=-1$ gives

$5(4)-3(7)=20-21=-1,$

so the point lies on line $l$. The vector equation also gives a corresponding point when the matching parameter value is used.

For coordinate geometry questions, the standard method is:

1. write $x$ and $y$ from the vector form,
2. substitute into the line equation,
3. solve for the parameter,
4. substitute back to get the coordinates.

Belangrijke methode

This is a classic example of solving an intersection problem with a parameter. The algebra is simple, but the structure matters: the vector form tells you how $x$ and $y$ depend on the parameter, and the Cartesian equation gives the condition that the point must satisfy. When both are combined correctly, the intersection point can be found directly.

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Pitfalls the pros know 👇 A common mistake is to stop after expanding the substitution and assume that the identity means there is no answer. In fact, an identity usually means the line equations describe the same line, or that the given working has a hidden inconsistency. Another frequent error is plugging the parameter into only one coordinate and forgetting to compute both $x$ and $y$. In vector-line questions, both coordinates must come from the same parameter value.

What if the problem changes? If line $l$ were changed to a different equation, such as $5x-3y=11$, the same substitution method would still apply, but the result would likely produce a single parameter value and one unique intersection point. If line $k$ were written in parametric form instead of vector form, for example $x=-2-3t$ and $y=-3-5t$, the solving process would be identical. A variant question might ask for the angle between the two lines after finding the intersection, which would require slope or direction-vector analysis.

Tags: vector equation of a line, intersection point, parametric coordinates