Is (7, 4) a solution of 3x – 4y = 5?
The Correct Answer and Explanation is:
Step 1: Substitute the values
3x−4y=53x – 4y = 53x−4y=53(7)−4(4)=21−16=53(7) – 4(4) = 21 – 16 = 53(7)−4(4)=21−16=55=55 = 55=5
Since both sides of the equation are equal, the equation is true when x=7x = 7x=7 and y=4y = 4y=4. This means that the point (7, 4) is a solution of the equation.
Explanation
An equation in two variables, such as 3x−4y=53x – 4y = 53x−4y=5, describes a relationship between the variables xxx and yyy. A solution of the equation is an ordered pair (x,y)(x, y)(x,y) that makes the equation true when the values are substituted into it. To determine whether a given point is a solution, substitute the xxx- and yyy-values from the point into the equation and simplify.
In this case, consider the point (7,4)(7, 4)(7,4) and the equation 3x−4y=53x – 4y = 53x−4y=5. Begin by substituting x=7x = 7x=7 and y=4y = 4y=4 into the left-hand side of the equation:3(7)−4(4)=21−16=53(7) – 4(4) = 21 – 16 = 53(7)−4(4)=21−16=5
After simplifying, the result is 5, which matches the right-hand side of the original equation. Since both sides of the equation are equal after substitution, the equation is satisfied.
Therefore, (7,4)(7, 4)(7,4) is a solution to the equation. This means that the point lies on the graph of the equation. In general, any point that satisfies the equation will lie on the line represented by that equation. If the substitution results in a false statement (like 5≠75 \ne 75=7), the point is not a solution and does not lie on the graph.
This method is useful for checking whether a specific point lies on the graph of a linear equation. Always substitute and simplify carefully to avoid errors.
