Question

Multiple Choice Which of the following is an equation of the line through \((-2,-1)\) parallel to the line \(y=-3 x+1 ?\) \(\begin{array}{ll}{\text { (A) } y=-3 x+5} & {\text { (B) } y=-3 x-7 \text { (C) } y=\frac{1}{3} x-\frac{1}{3}} \\ {\text { (D) } y=-3 x+1} & {\text { (E) } y=-3 x-4}\end{array}\)

Short answer

Option (B) \(y = -3x - 7\) is the correct answer, as it is the equation of the line through the point \((-2, -1)\) and parallel to the line \(y = -3x + 1\).

Step-by-step solution

Identify the slope

For the equation of the original line \(y = -3x + 1\), the slope is -3. Any line parallel to this one will also have the same slope, which is -3.

Substitute the given point and slope into the line equation

We have a point \((-2,-1)\) and we know that the slope (m) is -3. Now we substitute these into the equation of a line in the form: \(y = mx + b\). Here y is -1, m is -3, and x is -2. Therefore, we have \(-1 = -3*(-2) + b\). Simplifying this we can find the value of b.

Solve for b

Solving the equation \(-1 = 6 + b\), we obtain \(b = -1 - 6 = -7\).

Form the equation

Using the slope -3 and the y-intercept -7, the equation of our line is \(y = -3x - 7\).

Key concepts

Slope of a Line

The slope of a line in coordinate geometry is a measure of how steep the line is. It's often represented by the letter 'm', and it tells us how much the line rises (or falls) as we move along it, which is determined by the change in 'y' (vertical change) divided by the change in 'x' (horizontal change).

For any linear equation in the form of \( y = mx + b \), the coefficient 'm' is the slope. For example, in the equation \( y = -3x + 1 \), the slope is -3. This means for every one unit moved horizontally to the right, the line falls 3 units. In our exercise, we knew that a line parallel to this one would also have a slope of -3, because parallel lines always have the same slope.

Linear Equations

Linear equations form the basis of algebra and represent straight lines in coordinate geometry. The standard form for a linear equation is \( y = mx + b \), where 'm' represents the slope and 'b' the y-intercept, which is the point where the line crosses the y-axis.

These equations are immensely useful in determining the relationship between two variables. For example, when we know the slope and a point through which the line passes, we can find the equation of the line by plugging these values into the equation, as shown in our exercise solutions.

Y-Intercept

The y-intercept is a characteristic point of a line on a graph where the line crosses the y-axis. In the equation \( y = mx + b \), the 'b' represents the y-intercept. This value tells us the exact location on the y-axis where the line will be, regardless of the slope.

The y-intercept is especially important because, together with the slope, it fully determines the position of the line on the graph. In our exercise, after calculating the slope, we determined the y-intercept 'b' by substituting the x and y values of a point on the line into the linear equation and solving for 'b'.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This allows for a connection between algebraic equations and geometric curves. The traditional system uses a pair of axes perpendicular to each other, labeled 'x' and 'y', to define the position of points on the plane.

Using coordinate geometry, we can algebraically solve for distances, midpoints, slopes, and equations of lines among many other things. In the context of our exercise, we applied principles of coordinate geometry to find the equation of a line parallel to another line, passing through a given point.