Problem 49 - Intermediate Algebra

Question

How do we know that the graphs of \(2 x-3 y=6\) and \(-2 x+3 y=-6\) are the same line?

Step-by-Step Solution

Answer

The second equation is a scalar multiple of the first by -1, so they represent the same line.

Step 1: Write both equations in standard form

We start by examining whether both equations can be written in standard form \(Ax + By = C\). The first equation, \(2x - 3y = 6\), is already in standard form. Similarly, the second equation, \(-2x + 3y = -6\), is also in standard form.

Step 2: Compare coefficients

Next, look at the coefficients of the variables in both equations. In the first equation, the coefficients are \(A_1 = 2\), \(B_1 = -3\), and \(C_1 = 6\). For the second equation, the coefficients are \(A_2 = -2\), \(B_2 = 3\), and \(C_2 = -6\).

Step 3: Check for scalar multiplication

To determine if the equations represent the same line, one equation can be a scalar multiple of the other. Check if there exists some constant \(k\) such that \(A_2 = kA_1\), \(B_2 = kB_1\), and \(C_2 = kC_1\). For these equations, if we let \(k = -1\), then \(-2 = (-1) \times 2\), \(3 = (-1) \times (-3)\), and \(-6 = (-1) \times 6\).

Step 4: Confirm both equations represent the same line

Since we have shown that the second equation is a scalar multiple of the first by the factor \(k = -1\), both equations indeed represent the same line. This means they have identical sets of solutions and intersect at every point on the line, not just one.

Key Concepts

Scalar Multiplication

Scalar multiplication is an essential concept in algebra that helps you determine if two algebraic equations represent the same line. When we multiply each term of an equation by a constant, we change the scale but not the solution set of the equation. For the lines defined by the equations \(2x - 3y = 6\) and \(-2x + 3y = -6\), checking for scalar multiplication is key to knowing they are the same line.

To see if two equations are indeed scalar multiples, we need to verify if there is a constant, \(k\), such that each coefficient in one equation is \(k\) times its corresponding coefficient in the other equation. In our case:

• \(A_2 = -2\) is \(k\) times \(A_1 = 2\) (so \(-2 = -1 \times 2\))
• \(B_2 = 3\) is \(k\) times \(B_1 = -3\) (so \(3 = -1 \times -3\))
• \(C_2 = -6\) is \(k\) times \(C_1 = 6\) (so \(-6 = -1 \times 6\))

Thus, by choosing \(k = -1\), each coefficient of one equation is an inverse of the other. This makes the two equations represent the same line in the coordinate plane.

Standard Form of Linear Equations

The standard form of a linear equation is a specific way of writing linear equations that makes it straightforward to compare and analyze them. It is written as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers, and \(A\) and \(B\) are not both zero.

Both equations \(2x - 3y = 6\) and \(-2x + 3y = -6\) are already in standard form. This makes it easy to quickly assess their coefficients and apply algebraic techniques like comparison and scalar multiplication. The consistency of this form helps in understanding whether multiple equations could represent identical graphs by analyzing their coefficients directly.

Being able to organize an equation into standard form serves as a foundation for many advanced mathematical problem-solving techniques. It acts as a common language to describe linear relationships, which can be crucial for further applications in graphing or linear algebra.

Identical Lines in the Coordinate Plane

Determining if two algebraic equations describe identical lines on the coordinate plane is vital for understanding their relationships and interactions. Lines are considered identical if they overlay completely—their equations generate the same graph.

For the equations \(2x - 3y = 6\) and \(-2x + 3y = -6\), after using scalar multiplication, we establish that they describe the same line. This means that for every point \((x, y)\) that satisfies the first equation, it will also satisfy the second equation.

When two lines are identical, they do not just intersect at one or a few points; every single point lies on both lines. This results in any two points derived from the lines having identical coordinates, thereby generating a complete overlap when graphed. Using the core concepts of algebra, identifying identical lines simplifies many mathematical problems and is essential for solving systems of equations efficiently.