Problem 37 - Contemporary Precalculus

Question

Determine whether the point is on the graph of the given equation. $$(6,2) ; 3 y+x=12$$

Step-by-Step Solution

Answer

Answer: Yes, the point (6, 2) is on the graph of the given equation \(3y + x = 12\).

Step 1: Substitute the values of the point into the given equation

We are given the point \((6, 2)\) and we need to substitute the values of the point into the given equation: \(3y + x = 12\). So we will substitute \(x=6\) and \(y=2\): $$3(2) + 6 = 12$$

Step 2: Check if the equation holds true

Now we need to check if substituting the point values in the equation holds true. We currently have the equation: $$3(2) + 6 = 12$$ Now, calculate the left side of the equation: $$6 + 6 = 12$$ Since \(12=12\), the equation holds true when we substitute the values of the point \((6, 2)\) into the given equation \(3y+x=12\).

Step 3: Conclusion

Since the equation holds true after substituting the values of the point \((6, 2)\), we can conclude that the point \((6, 2)\) is indeed on the graph of the given equation \(3y+x=12\).

Key Concepts

Substituting Values in Equations

Understanding how to substitute values into equations is fundamental for solving algebraic problems. In the context of graphs and functions, when we are given a point, like (6,2), we are actually dealing with a set of coordinates. The first number represents the x-coordinate, while the second number is the y-coordinate.

Now, to verify if this point lies on the graph of an equation, such as \(3y + x = 12\), we must replace \(x\) with 6 and \(y\) with 2. The process involves plugging these coordinates into the equation:\[3(2) + 6 = 12\]After simplification, we obtain \(6 + 6 = 12\), which verifies that the point (6,2) satisfies the equation. This substitution technique is not limited to verifying points; it's a cornerstone method used to solve various types of algebraic equations.

Solution of Linear Equations

Linear equations form the backbone of algebra and represent relationships where variables change at a constant rate. They are equations of the first degree, meaning that the variable(s) are not raised to any power other than 1.

A standard form for a linear equation in two variables, x and y, looks like \(Ax + By = C\), where A, B, and C are constants. To find a solution to a linear equation means determining a set of values for the variables that make the equation true. In our case, we are looking at the equation \(3y + x = 12\), which is already a solvable linear equation.

To solve for y when x is known (or vice versa), we rearrange the terms: \[y = \frac{12 - x}{3}\]Finding a solution involves substituting a value for x and calculating y, or substituting a value for y and calculating x. Any point that satisfies the equation represents a solution and is a point on the graph of the equation.

Verification of Points in Equations

Verification is a crucial step in mathematics to ensure our solutions are accurate. To verify a point in the context of an equation means to prove whether the point lies on the line or curve that the equation represents in a graph.

To verify a point, we substitute the x and y coordinates into the equation and simplify the expression. If both sides of the equation balance — they produce the same number — then the point is verified and it lies on the graph. In case of our example, by substituting (6, 2) into \(3y + x = 12\), we get \(12 = 12\), which confirms our point lies on the line represented by the equation.

Verification is not only a step to confirm a solution but also helps us understand the relationship represented by the equation. Each verified point serves as a puzzle piece that visually builds up the graph associated with the equation.