91Ó°ÊÓ

To find the solutions to a linear equation, you start by choosing values for one of the variables to make the equation easier to solve. Let's focus on the equation from the exercise: \( y = -4x + 5 \).

First, choose a value for \( x \). This value can be any number. For example, if we select \( x = 0 \), we then substitute this into the equation:

• \( y = -4(0) + 5 \) simplifies to \( y = 5 \).

This gives us one solution: \( (0, 5) \).

Repeat this process again with different \( x \) values to find more solutions. For example, choosing \( x = 1 \):

• \( y = -4(1) + 5 \) simplifies to \( y = 1 \).

And choosing \( x = -1 \):

• \( y = -4(-1) + 5 \) simplifies to \( y = 9 \).

So our additional solutions are \( (1, 1) \) and \( (-1, 9) \).

By choosing various values for \( x \), you can find multiple pairs of \( (x, y) \) that satisfy the equation, giving you several solutions.

The substitution method is a powerful technique used to find solutions of linear equations. It involves replacing one variable with a value or another expression.

Here's how it works using our example: \( y = -4x + 5 \).

First, we pick a value for \( x \). If we select \( x = 0 \), we substitute 0 for \( x \) in the equation:

• \( y = -4(0) + 5 \)

This simplifies to \( y = 5 \). Therefore, one solution is \( (0, 5) \).

Next, choose another \( x \) value, like \( x = 1 \), and substitute it:

• \( y = -4(1) + 5 \)

This simplifies to \( y = 1 \).
Another solution is \( (1, 1) \).

Repeat this process. If you choose \( x = -1 \) and substitute:

• \( y = -4(-1) + 5 \)

This simplifies to \( y = 9 \).
So the third solution is \( (-1, 9) \).

By systematically substituting different values of \( x \) into the equation, you can easily solve for \( y \).

Coordinate pairs represent solutions to equations in a graphical form, where each pair \((x, y)\) corresponds to a point on a graph.

For the equation \( y = -4x + 5 \), each solution we find is a coordinate pair:

• (0, 5)
• (1, 1)
• (-1, 9)

These pairs show exactly where the line of the equation crosses the \( x \) and \( y \) axes.

Plotting these pairs on a graph can help visualize the solutions and better understand the equation. Each solution \((x, y)\) represents a point where the line passes through the coordinate plane. For example:

• (0, 5) means that at \( x = 0 \), \( y = 5 \)
• (1, 1) means that at \( x = 1 \), \( y = 1 \)
• (-1, 9) means that at \( x = -1 \), \( y = 9 \)

Seeing these points on a graph can solidify your understanding of how linear equations behave. As more points are plotted, you can see the straight line formed by the equation's solutions.