Problem 111
Question
Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The graph of the linear function \(5 x+6 y-30=0\) is a line passing through the point \((6,0)\) with slope \(-\frac{5}{6}\).
Step-by-Step Solution
Verified Answer
The original statement 'The graph of the linear function \(5x + 6y - 30 = 0\) is a line passing through the point \((6,0)\) with slope -\frac{5}{6}' is correct. Therefore, no changes are required.
1Step 1: Equation in Standard Form
Rewrite the equation \( 5x + 6y - 30 = 0 \) into the standard form of a line equation i.e., \( y = mx + c \). After rewriting, the equation would become \(y = -\frac{5}{6}x + 5\).
2Step 2: Verify the Point
Substitute the point \((6,0)\) into the rewritten equation. After substitution, you obtain \(0 = -\frac{5}{6} * 6 + 5\). Simplifying it yields \(0 = 0\). So, the line does pass through the point (6,0).
3Step 3: Verify the Slope
Slope of the line from the given equation in standard form \(y = mx + c\) is -\frac{5}{6}. This is the same slope as mentioned in the statement.
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