Determining Solutions to Linear Equations
Hello Math Students! Today, we focused on determining whether a point is a solution to a given linear equation. This involves substituting the $x$ and $y$ coordinates of the point into the equation and checking if the equation holds true. Let's break down the process with some examples.
Key Concept: A point $(x, y)$ is a solution to a linear equation if, after substituting the values of $x$ and $y$ into the equation, both sides of the equation are equal.
How to Check if a Point is a Solution:
- Identify the Equation and the Point: Start with the linear equation and the coordinate point $(x, y)$ you want to test.
- Substitute: Replace $x$ and $y$ in the equation with the $x$ and $y$ values from the given point.
- Simplify: Perform the necessary arithmetic operations on both sides of the equation.
- Verify: Check if both sides of the equation are equal. If they are, the point is a solution. If not, the point is not a solution.
Examples:
Example 1:
Let's determine if the point $(3, 9)$ is a solution to the equation $y = 2x + 3$.
Substitute $x = 3$ and $y = 9$ into the equation:
$$9 = 2(3) + 3$$
Simplify:
$$9 = 6 + 3$$
$$9 = 9$$
Since both sides are equal, the point $(3, 9)$ is a solution to the equation $y = 2x + 3$.
Example 2:
Let's determine if the point $(-5, -6)$ is a solution to the equation $y = 2x + 3$.
Substitute $x = -5$ and $y = -6$ into the equation:
$$-6 = 2(-5) + 3$$
Simplify:
$$-6 = -10 + 3$$
$$-6 = -7$$
Since both sides are not equal, the point $(-5, -6)$ is not a solution to the equation $y = 2x + 3$.
Example 3:
Let's determine if the point $(3, -7)$ is a solution to the equation $y = -x - 4$.
Substitute $x = 3$ and $y = -7$ into the equation:
$$-7 = -(3) - 4$$
Simplify:
$$-7 = -3 - 4$$
$$-7 = -7$$
Since both sides are equal, the point $(3, -7)$ is a solution to the equation $y = -x - 4$.
Homework:
Complete the exercises on Pg. 142, problems #12-18 even. Practice makes perfect! Make sure to show all of your steps. Good luck, and see you in the next class!