Analyzing the Relationship Between Linear Equations

We are given two linear equations:

1. Equation 1: $2x - y = 3$
2. Equation 2: $4x - y = 5$

To determine the relationship between the lines represented by these equations (whether they intersect, are parallel, or are coincident), we can compare the ratios of their coefficients. A standard form for a linear equation is $Ax + By = C$.

For Equation 1 ($2x - y = 3$): $A_1 = 2$, $B_1 = -1$, $C_1 = 3$.

For Equation 2 ($4x - y = 5$): $A_2 = 4$, $B_2 = -1$, $C_2 = 5$.

We compare the ratios $\frac{A_1}{A_2}$, $\frac{B_1}{B_2}$, and $\frac{C_1}{C_2}$.

• If $\frac{A_1}{A_2} \ne \frac{B_1}{B_2}$, the lines intersect at a single point.
• If $\frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2}$, the lines are coincident (they are the same line).
• If $\frac{A_1}{A_2} = \frac{B_1}{B_2} \ne \frac{C_1}{C_2}$, the lines are parallel and distinct.

Let's calculate the ratios for our given equations:

• Ratio of x-coefficients: $\frac{A_1}{A_2} = \frac{2}{4} = \frac{1}{2}$
• Ratio of y-coefficients: $\frac{B_1}{B_2} = \frac{-1}{-1} = 1$
• Ratio of constant terms: $\frac{C_1}{C_2} = \frac{3}{5}$

Now, let's compare these ratios:

Is $\frac{A_1}{A_2} = \frac{B_1}{B_2}$?

$\frac{1}{2} = 1$ is false.

Since $\frac{A_1}{A_2} \ne \frac{B_1}{B_2}$ (specifically, $\frac{1}{2} \ne 1$), the condition for intersecting lines is met.

This means the lines representing the linear equations $2x - y = 3$ and $4x - y = 5$ intersect at exactly one point.

Alternative Method: Solving the System of Linear Equations

We can also solve the system of equations to find the intersection point. If a unique solution exists, the lines intersect at one point.

Equation 1: $2x - y = 3$

Equation 2: $4x - y = 5$

We can use the elimination method. Subtract Equation 1 from Equation 2:

$(4x - y) - (2x - y) = 5 - 3$

$4x - y - 2x + y = 2$

$2x = 2$

$x = \frac{2}{2} = 1$

Now substitute the value of $x$ into Equation 1:

$2(1) - y = 3$

$2 - y = 3$

$-y = 3 - 2$

$-y = 1$

$y = -1$

The solution to the system is $(x, y) = (1, -1)$. Since there is a unique solution, the lines intersect at the point $(1, -1)$.

Conclusion on Linear Equations' Relationship

Based on the analysis of the coefficient ratios and the solution of the system, the lines representing the linear equations $2x - y = 3$ and $4x - y = 5$ intersect at a single point.

Revision Table: Types of Linear Systems

Additional Information on Linear Equation Systems

A system of linear equations is a set of two or more linear equations involving the same variables. The solution to a system of linear equations is the point or set of points that satisfy all equations in the system simultaneously. Graphically, this corresponds to the intersection points of the lines (or planes, in higher dimensions) represented by the equations.

Linear equations in two variables ($x$ and $y$) represent straight lines when plotted on a coordinate plane. The relationship between these lines provides insight into the number of solutions the system has.

• Consistent System: A system of linear equations is consistent if it has at least one solution. This occurs when the lines are intersecting or coincident.
• Inconsistent System: A system of linear equations is inconsistent if it has no solution. This occurs when the lines are parallel and distinct.
• Independent System: A consistent system is independent if it has exactly one solution (intersecting lines).
• Dependent System: A consistent system is dependent if it has infinitely many solutions (coincident lines).

In our specific case, the system of linear equations $2x - y = 3$ and $4x - y = 5$ is a consistent and independent system because the lines intersect at a single point, $(1, -1)$.