Problem 58 - Algebra 1

Question

LOGICAL REASONING In Exercises \(56-59\), tell whether the statement is true or false. Justify your answer. The point \((3,5)\) is on the graph of \(3 x+5 y=30\)

Step-by-Step Solution

Answer

The statement is false. The point \((3,5)\) is not on the graph of \(3 x + 5 y = 30\) since on substituting the point into the equality, both sides of the equation are not equal.

Step 1: Substitute the point into the equation

Replace \(x\) in the equation with 3 and \(y\) with 5.\n It becomes: \(3*3 + 5*5\).

Step 2: Compute the equation

Calculate the left side of the equation. \(3*3 + 5*5 = 9 + 25 = 34\).\n Now, compare this with the right side of the equation, which is 30.

Step 3: Comparing both sides

As 34 is not equal to 30. The left hand side is not equal to the right hand side of the equation after substituting the given point into it. Therefore, the point \((3,5)\) does not lie on the graph of the given equation.

Key Concepts

Graphing Linear Equations

Graphing linear equations involves plotting a straight line on a coordinate system. This linear equation can be represented in the form: \(ax + by = c\). The values are coefficients of \(x\) and \(y\), and the constant \(c\).
To graph this,

• Choose values for \(x\) to find corresponding \(y\)
• Plot several points \((x, y)\)
• Connect these points to form the line

Lines are characterized by having consistent slope, calculated as the change in \(y\) divided by the change in \(x\). This is crucial for understanding how the slope will influence the position and angle of your line on the graph. Consider identifying at least two main points—the x-intercept, where the line crosses the x-axis, and the y-intercept, where it crosses the y-axis, to accurately draw the line.

Point on a Line

A key concept in linear algebra is determining whether a specific point lies on a line defined by an equation. To verify,

• Insert the point's coordinates into the equation
• Check if both sides of the equation match

For instance, given the linear equation \(3x + 5y = 30\) and a point \((3, 5)\), substitute \(x = 3\) and \(y = 5\) into the equation:
\(3(3) + 5(5) = 9 + 25 = 34\).
Since the computed value is not equal to the constant on the right side of the equation (30), the point does not lie on this line.
This method applies to any linear equation, showing whether or not a specified point falls on the graphed line, providing a foundation for systematically exploring relationships between variables.

Substitution Method

The substitution method is a valuable algebraic technique used to solve equations and confirm hypotheses about graphs and points. It involves replacing one variable with its equivalent from another context to simplify and solve equations.
To use it:

• Select a value or expression for a variable
• Insert this expression into another equation in place of that variable
• Solve the resulting equation

This strategy is particularly handy in systems of equations where you have multiple variables, as it reduces complexity by focusing on one variable at a time. It also aids in verifying if a point is on a line, like in our earlier example of testing if \((3, 5)\) is on \(3x + 5y = 30\).
By substituting \(x=3\) and \(y=5\), calculation showed \(3(3) + 5(5)\) did not satisfy \(30\). Thus validating the point does not stand on the line, illustrating how the substitution method clarifies mathematical relationships succinctly and effectively.