Problem 22 - Algebra 1: Concepts and Skills

Question

Find three ordered pairs that are solutions of the equation. $$ x=-10 $$

Step-by-Step Solution

Answer

The three ordered pairs that satisfy the equation \(x=-10\) are (-10, -1), (-10, 0) and (-10, 1).

Step 1 Prepare the values of x

The equation is \(x=-10\), meaning for all the solutions of the equation, x will always be -10. So, it's already given what 'x' is.

Step 2 Select any three values for y

Since there's no 'y' in our equation, we can choose any three values for 'y'. Those can be any real numbers. For simplicity, let's choose -1, 0 and 1.

Step 3 Form the ordered pairs

Now that we have chosen our 'y' values, we can form ordered pairs (x,y) that satisfy the equation \(x=-10\). With 'x' being -10 and 'y' taking values -1, 0, 1 - we get the pairs (-10, -1), (-10, 0) and (-10, 1). These are the three pairs that satisfy the equation.

Key Concepts

Linear Equations

Linear equations are a fundamental concept in algebra that describe relationships between variables. They can be represented as lines on a graph. In most cases, a linear equation appears in the form of \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
However, sometimes you encounter linear equations like \(x = -10\), where there is no \(y\) component. Here, the equation signifies that no matter what the \(y\) value is, the \(x\) will always be -10. This makes the line vertical in orientation.

• Key Point: Linear equations can represent either horizontal, vertical, or slanted lines.
• Vertical lines, like \(x = -10\), indicate a constant \(x\) value and are parallel to the y-axis.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface created by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). These axes intersect at the origin, with coordinates (0, 0). The plane is divided into four quadrants, determined by the positive and negative values of x and y.
When working with a linear equation like \(x = -10\), we plot points on this plane by finding all possible ordered pairs with \(x\) as -10. This tells us every possible position on the line where the x-coordinate remains constant.

• Ordered pairs are written as (x, y) and represent points on the coordinate plane.
• Vertical lines, such as \(x = -10\), run upward and downward across the plane.

Solutions of Equations

Solutions of linear equations are the ordered pairs (x, y) that make the equation true. For the equation \(x = -10\), finding solutions involves choosing any value for \(y\) because \(x\) is already established. This makes it straightforward to determine solutions since only one component, \(x\), is restricted.
In our example, we selected \(y = -1, 0, 1\) to easily identify three solutions: (-10, -1), (-10, 0), and (-10, 1).

• A solution is valid if it satisfies the equation when substituted back into it.
• The infinite number of possible values for \(y\) means there are infinitely many solutions for the equation \(x = -10\).