Problem 37 - Algebra 1: Concepts and Skills

Question

Write in standard form an equation of the line that passes through the two points. Use integer coefficients. \((-3,3),(7,2)\)

Step-by-Step Solution

Answer

The standard form equation of the line that passes through the points (-3,3) and (7,2) is \(x + 10y = 27\).

Step 1: Find the slope

Firstly, we calculate the slope of the line (m) using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Replace \(x_1, y_1\) with the coordinates of the first point (-3,3) and \(x_2, y_2\) with the coordinates of the second point (7,2). So \(m = \frac{2 - 3}{7 - (-3)} = -\frac{1}{10}\).

Step 2: Substitute one point and slope into point-slope equation of line

Now that we have the slope, insert the slope and one of the points in the point-slope form of the line equation \(y - y_1 = m(x - x_1)\). Let us use the first point (-3,3), which gives us \(y - 3 = -\frac{1}{10}(x - (-3))\)

Step 3: Transform into standard form

Standard form for a line is defined as \(Ax + By = C\) where A, B, and C are integers. We transform the equation from step 2 into standard form by multiplying every term by 10 to eliminate fractions: \(10(y - 3) = -1(x + 3)\) so \(10y - 30 = -x - 3\) and finally rearranging to \(x +10y = 27\).

Key Concepts

Standard Form

The standard form of a linear equation is a common way to represent a line. It is expressed as:

• \( Ax + By = C \)

where \(A\), \(B\), and \(C\) are integers, and \(A\) and \(B\) are not both zero. The coefficients \(A\) and \(B\) define the line's orientation on the coordinate plane.
This form is particularly useful because it clearly shows the coefficients and the constant term. It allows for easy determination of intercepts and can define vertical lines efficiently when \(B = 0\).
In our exercise, the line that passes through \((-3, 3)\) and \((7, 2)\) was converted to the standard form \(x + 10y = 27\). This was achieved by first determining the slope and using point-slope form to finally rearrange into standard form.

Slope

The slope of a line is a measure of its steepness and direction. Mathematically, slope \(m\) is defined as the "rise" over the "run," or the change in the \(y\)-coordinates divided by the change in the \(x\)-coordinates between two points.The formula for calculating the slope \(m\) is:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

In our example, using points \((-3, 3)\) and \((7, 2)\), the slope was calculated to be \(-\frac{1}{10}\).
This tells us that for every 10 units you move horizontally from left to right, the line moves down 1 unit vertically. Slope is essential for understanding the angle and direction of a line on a plane.

Point-Slope Form

Point-slope form is a handy method for writing the equation of a line. It is especially useful if you know one point on the line and the slope. The general formula for point-slope form is:

• \( y - y_1 = m(x - x_1) \)

Here, \(m\) represents the slope, and \((x_1, y_1)\) is a specific point on the line.
In the exercise, after finding the slope \(m = -\frac{1}{10}\) of the line through points \((-3, 3)\) and \((7, 2)\), the point-slope form becomes \( y - 3 = -\frac{1}{10}(x + 3) \). This version of the line highlights how we move from a point using the slope to define every other point on the line.

Integer Coefficients

Integer coefficients in the context of linear equations mean that all the coefficients \(A\), \(B\), and \(C\) in the standard form are whole numbers – no fractions or decimals.Having integer coefficients makes equations cleaner and easier to work with. In many applications, especially where the equation is to be visually interpreted or checked by others, integer coefficients provide clarity.In the solution, converting the point-slope equation to standard form involved eliminating the fraction by multiplying through to clear any non-integer numbers. This gives us the final equation \(x + 10y = 27\). Ensuring integer coefficients often involves steps like these to transform the equation from one form to another.