Problem 13 - Algebra 1

Question

Write an equation in standard form of the line that passes through the given point and has the given slope. $$(5,-8), m=-1$$

Step-by-Step Solution

Answer

The equation in standard form of the line that passes through the point (5,-8) and has the slope -1 is \(x + y = -3\).

Step 1: Insert given values into point-slope formula

First, we substitute point (5,-8) and slope -1 into the point-slope formula to find a line equation. It will be \(y + 8 = -1(x - 5)\).

Step 2: Simplify the equation

Next, distribute the -1 to the terms inside the parentheses to give us: \(y + 8 = -x + 5\).

Step 3: Transpose to Standard Form

Finally, we can re-arrange the equation to the form Ax + By = C. Adding x to both sides, we obtain \(x + y = -8 + 5\) which simplifies to \(x + y = -3\).

Key Concepts

Standard Form of a Line

Understanding the standard form of a line is crucial when dealing with linear equations. The standard form is typically written as
\( Ax + By = C \), where
\( A \),
\( B \), and
\( C \) are integers, and
\( A > 0 \). An equation in this form is particularly useful for certain algebraic operations such as finding intercepts and is advantageous in systems of equations.

• For clarity:
  • \( A \), \( B \) are the coefficients of \( x \) and \( y \), respectively,
  • \( C \) represents the constant term.
• The exercise improvement advice here suggests ensuring the coefficient \( A \) is positive when writing the final equation.
• If any fraction coefficients exist, they should be eliminated.
• The coefficients should be the simplest integer values, which can be achieved by finding the greatest common divisor (GCD).

For the given problem, the student began with an equation and successfully reached the standard form \( x + y = -3 \), in line with the characteristics of the standard form stated above.

Point-Slope Formula

The point-slope formula is a straightforward method for writing the equation of a line when you have a point on the line \((x_1, y_1)\) and the slope \(m\). This formula is given by
\( y - y_1 = m(x - x_1) \).

• This formula creates an equation that shows the relationship between any point \((x, y)\) on the line.
• The slope \(m\) indicates the steepness and direction of the line.
• Point \((x_1, y_1)\) is plugged directly into the formula along with the slope value.

The power of the point-slope formula lies in its simplicity and direct approach. From the homework example, using the point \((5, -8)\) and slope \(-1\), the formula quickly allows for the construction of an equation that leads to the line's standard form.
Though simple, careful algebraic manipulation, as shown in the exercise, is necessary to then convert this into the standard form.

Linear Equations

Linear equations form the backbone of algebra and represent straight lines on a graph. Such an equation involves variables raised to the power of one, and it describes a constant rate of change, or slope. Major forms of linear equations include:

• Slope-intercept form: This is written as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept,
• Point-slope form: Discussed in the previous section and ideal for when a point and a slope are known,
• Standard form: As detailed earlier.

These different forms can be transformed into one another with algebraic manipulation. For instance, to turn the point-slope form into the standard form, terms are rearranged and sometimes coefficients are multiplied or divided to achieve integer values. Acknowledging these various representations is important for solving various algebraic problems, as each form may offer different benefits depending on the scenario at hand.