- All three equations represent the same line, as they all simplify to the slope-intercept form \(y = 3x + 15\).
- The line has a slope of \(m = 3\).
- The line passes through the y-intercept at \((0, 15)\) and the x-intercept at \((-5, 0)\).
- The line can be visualized by connecting the points \((0, 15)\) and \((-5, 0)\) on a 2D plane.
- The finalized line is given by \(\boxed{y = 3x + 15}\).
Explanation
- Confirming the Equations Represent the Same Line
We are given three linear equations:
A\(6x - 2y = -30\)
B\(y = 3x + 15\)
C\(y - 18 = 3(x - 1)\)
We first need to confirm that these equations all represent the same line and extract key information such as the slope and intercepts. Converting each equation to the slope-intercept form (\(y = mx + b\)) is an effective way to compare them. For equation (a), \(6x - 2y = -30\), we subtract \(6x\) from both sides to get \(-2y = -6x - 30\). Dividing by \(-2\) yields \(y = 3x + 15\). Equation (b) is already given in this form. For equation (c), \(y - 18 = 3(x - 1)\), distributing the \(3\) gives \(y - 18 = 3x - 3\), and adding \(18\) to both sides results in \(y = 3x + 15\). All three expressions simplify to the same slope-intercept form, confirming they describe the identical line.
- Identifying Key Line Characteristics
From the slope-intercept form \(y = 3x + 15\), we can identify two essential characteristics of the line:
- The slope \(m = 3\), which tells us the line rises 3 units vertically for every 1 unit of horizontal movement to the right.
- The y-intercept is found by setting \(x = 0\), giving \(y = 15\), so the line passes through \((0, 15)\).
- The x-intercept is found by setting \(y = 0\), resulting in \(0 = 3x + 15\), which solves to \(3x = -15\), or \(x = -5\), meaning the line passes through \((-5, 0)\).
- Graphing the Line
To graph the line, we plot the two points we identified: the y-intercept at \((0, 15)\) and the x-intercept at \((-5, 0)\). By drawing a straight line passing through these two coordinates on a Cartesian coordinate system, we represent the set of all points \((x, y)\) that satisfy the equation. This linear graph shows a constant positive slope, confirming the relationship defined by the equations.
Examples
In urban planning, identifying equivalent expressions for linear paths is crucial. For instance, if a proposed road is defined by different constraints—such as its starting coordinate, its steepness (slope), or a specific point it must cross—civil engineers use these equivalent forms to ensure all designs align perfectly. Recognizing that \(6x - 2y = -30\) and \(y = 3x + 15\) describe the same path ensures that different departments working with different specifications are building the same road.