When working with linear equations in two variables, such as the equation y = 4x + 5, it is important to understand how to find a solution to the equation. In this article, we will explore how to determine which point is a solution to the equation y = 4x + 5. We will discuss the concept of solutions, how to find solutions graphically and algebraically, and provide examples to illustrate these concepts.
The Concept of Solutions
Before we delve into finding solutions to the equation y = 4x + 5, let’s first understand the concept of solutions in the context of linear equations. A solution to a linear equation in two variables is a pair of values (x, y) that make the equation true. In other words, when you substitute the values of x and y into the equation, the equation will be satisfied.
Finding Solutions Graphically
One way to find a solution to the equation y = 4x + 5 is by graphing the equation on a coordinate plane. By graphing the equation, the solution will be the point where the graph intersects the line y = 4x + 5. Here are the steps to find the solution graphically:
- Step 1: Plot the y-intercept: To plot the y-intercept, you need to find the point where the line intersects the y-axis. In this case, the y-intercept is 5, so plot the point (0, 5).
- Step 2: Find another point on the line: To find another point on the line, you can choose any value of x and substitute it into the equation y = 4x + 5. For example, if x = 1, then y = 4 * 1 + 5 = 9. So, plot the point (1, 9).
- Step 3: Draw a line through the two points: Once you have plotted two points on the line, draw a straight line that passes through both points. This line represents the equation y = 4x + 5.
- Step 4: Identify the solution: The solution to the equation y = 4x + 5 is the point where the graph intersects the line y = 4x + 5. This point is the solution to the equation.
Finding Solutions Algebraically
Another way to find a solution to the equation y = 4x + 5 is by solving the equation algebraically. By substituting the values of x and y into the equation, you can determine which point is a solution. Here are the steps to find the solution algebraically:
- Step 1: Substitute the values of x and y: Let’s say you have a point (x, y) that you want to check if it is a solution to the equation y = 4x + 5. Substitute the values of x and y into the equation: y = 4x + 5.
- Step 2: Simplify the equation: After substituting the values of x and y, simplify the equation to see if both sides are equal. If they are equal, then the point is a solution to the equation.
- Step 3: Check for consistency: Make sure to check if the values satisfy the equation consistently. If the values of x and y satisfy the equation consistently, then the point is a solution.
Example Problems
Let’s work through a couple of examples to illustrate how to find a solution to the equation y = 4x + 5:
Example 1:
Find out which point is a solution to the equation y = 4x + 5 for the point (2, 13).
Solution:
- Substitute x = 2 and y = 13 into the equation y = 4x + 5.
- 13 = 4 * 2 + 5
- 13 = 8 + 5
- 13 = 13
Since both sides of the equation are equal, the point (2, 13) is a solution to the equation y = 4x + 5.
Example 2:
Determine which point is a solution to the equation y = 4x + 5 for the point (-3, -7).
Solution:
- Substitute x = -3 and y = -7 into the equation y = 4x + 5.
- -7 = 4 * (-3) + 5
- -7 = -12 + 5
- -7 = -7
Since both sides of the equation are equal, the point (-3, -7) is a solution to the equation y = 4x + 5.
Conclusion
In conclusion, finding a solution to the equation y = 4x + 5 involves determining which point satisfies the equation. Whether you find the solution graphically by plotting the equation on a coordinate plane or algebraically by substituting the values of x and y into the equation, the key is to ensure consistency between both sides of the equation. By following the steps outlined in this article and practicing with examples, you can confidently identify which point is a solution to the equation y = 4x + 5.
