Problem 41 - Algebra 1

Question

Write an equation in standard form of the line that passes through the given point and has the given slope. $$(7,3), m=-2$$

Step-by-Step Solution

Answer

The equation of the line in standard form is \(2x + y = 17\).

Step 1: Write in Point-Slope Form

The point-slope form of a line's equation is given by \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is a given point the line passes through. In this case, substituting \((7,3)\) for \((x_1,y_1)\) and \(-2\) for \(m\), we get \(y - 3 = -2(x - 7)\).

Step 2: Simplify the Equation

Now simplify the equation. Distribute \(-2\) to the terms inside the parenthesis to obtain \(y - 3 = -2x + 14\).

Step 3: Convert to Standard Form

The standard form of a line's equation is \(Ax + By = C\). We must convert our equation into this form. By moving \(2x\) from right to left and \(3\) from left to right, we obtain \(2x + y = 17\).

Key Concepts

Point-Slope Form

The point-slope form is one of the ways to express the equation of a line. It is particularly useful when you have one point on the line and the slope. The formula is given by:\[y - y_1 = m(x - x_1)\]where:

• \((x_1, y_1)\) is a point on the line.
• \(m\) is the slope of the line.

In our example, the point given is \((7, 3)\) and the slope is \(-2\). By plugging these values into the formula, we arrive at:\[y - 3 = -2(x - 7)\]This form makes it straightforward to visualize how the line behaves in terms of rise and run from a specific point.
It emphasizes understanding how the slope affects changes in \(y\) with respect to \(x\) from a known location.To move forward to solving, this form serves as the launching pad for simplifying and converting into other forms, like the standard form.

Standard Form

The standard form of a linear equation is expressed as \(Ax + By = C\). Here, \(A\), \(B\), and \(C\) are integers, and ideally, this form does not have fractions.
This form is preferred for quickly identifying intersections with the axes, as well as adding or comparing multiple linear equations efficiently. When converting from point-slope to standard form, the aim is to rearrange the equation so all variables and constants are on one side.In the exercise, after writing the equation in point-slope form, it was converted:After the step with \(y - 3 = -2x + 14\), move \(-2x\) to the left and \(3\) to the right. This results in:\[2x + y = 17\]This transformation gives a clear, balanced visual of how \(x\) and \(y\) interact to form the line, beneficial for solving systems of equations effectively.

Coordinate Geometry

Coordinate geometry, often referred to as analytic geometry, uses algebraic equations to represent and solve geometric problems on a coordinate plane.
This powerful tool connects algebraic expressions with geometric figures by representing points, lines, and shapes using a set of equals.The core of coordinate geometry lies in using coordinates \((x, y)\) to locate points on a plane.
For lines, understanding how forms such as point-slope and standard form offer insights into geometric relationships is crucial.For a line with a slope like \(-2\) in our example, it means every step "right" along the \(x\)-axis, the line "falls" 2 steps on the \(y\)-axis.
This beautifully aligns with visualizing graphs and solving geometric problems systematically using algebra.In the scope of coordinate geometry, transforming equations between different forms enriches understanding and allows us to tackle a variety of problems, such as finding intersections, distances, and more on the coordinate plane.