Identify the slope ($m$) and $y$-intercept ($b$) for each of the following:
Final Answer Summary:
The slope-intercept form of a linear equation is given by \(y = mx + b\), where \(m\) represents the slope (the rate of change of \(y\) with respect to \(x\)) and \(b\) represents the \(y\)-intercept (the value of \(y\) when \(x=0\), corresponding to the point \((0, b)\)). We will analyze each case provided to identify these parameters.
For the equation \(y = x + 7\), we can rewrite this as \(y = 1x + 7\). By comparing this with the standard form \(y = mx + b\), we clearly see that the coefficient of \(x\) is 1 and the constant term is 7. Therefore, the slope is \(m = 1\) and the \(y\)-intercept is \(b = 7\).
For the equation \(y = 9\), this is a constant function. We can express it as \(y = 0x + 9\). Comparing this with \(y = mx + b\), the coefficient of \(x\) is 0 and the constant term is 9. Thus, the slope is \(m = 0\) and the \(y\)-intercept is \(b = 9\). This represents a horizontal line passing through the point \((0, 9)\).
The values for part c are provided as \(m = -\frac{2}{2}\) and \(b = -\frac{4}{1}\). Simplifying these fractions, we find \(m = -1\) and \(b = -4\). If this corresponds to an equation, it would be \(y = -x - 4\).
Part d provides placeholders for \(m\) and \(b\) but no specific equation or values. Without additional information or an equation to reference, the slope and \(y\)-intercept cannot be determined.
Understanding slopes and intercepts is essential in real-life applications. For example, if a taxi service charges a base fee of \(\$7\) (the y-intercept \(b\)) plus \(\$1\) for every kilometer traveled (the slope \(m\)), the cost \(y\) for traveling \(x\) kilometers is represented by \(y = 1x + 7\). Knowing these parameters helps in calculating the total fare for any trip length.