Question

complete the following equations. enter an equation for line t: y = enter an equation for line a: y = line a is a translation of line t. writing equations for lines

Explanation:

Step1: Find the slope - intercept form of a line

The equation of a line is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. For parallel lines (a translation of a line has the same slope as the original line), if we assume the slope of line \(t\) and line \(a\) is \(m\). First, we need to find the slope of line \(t\) using two points on line \(t\). Let's assume two points \((x_1,y_1)\) and \((x_2,y_2)\) on line \(t\). The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\).

Step2: Find the y - intercept of line \(a\)

If we know the slope \(m\) and a point \((x_0,y_0)\) on line \(a\), we can substitute into the equation \(y=mx + b\) to find \(b\). \(b=y_0 - mx_0\).
However, since we don't have the actual points' coordinates clearly visible in the image for a full - fledged calculation, if we assume the slope of line \(t\) is \(m=- 1\) (from the given partial equation of line \(t:y=-1x\)). Since line \(a\) is a translation of line \(t\), the slope of line \(a\) is also \(m=-1\). Let's assume a point \((x_1,y_1)\) on line \(a\) is \((0, - 3)\) (by visual inspection, assuming the y - intercept of line \(a\)).
The equation of line \(a\) in slope - intercept form \(y=-1x+( - 3)\)

Answer:

For line \(a:y=-1x+( - 3)\)
For line \(t\) (assuming the given partial information is correct): \(y=-1x\) (it seems the y - intercept of line \(t\) is \(0\) from the given \(y = - 1x\) format)