Problem 9 - Algebra 1: Concepts and Skills

Question

Write in standard form an equation of the line that passes through the two points. Use integer coefficients. \((3,1),(4,-2)\)

Step-by-Step Solution

Answer

The line that passes through the points (3,1) and (4,-2) can be described by the standard form equation 3x+y = 10.

Step 1: Calculate the Slope

We first calculate the slope (m) of the line. The formula for slope between two points A(x_1, y_1) and B(x_2, y_2) is \[m = \frac{y_2 - y_1}{x_2 - x_1}\]. Substituting the values of the given points (3,1) and (4,-2) into the formula, we get: \[m = \frac{-2 - 1}{4 - 3} = -3\]

Step 2: Apply Point-Slope Formula

Next, we take the slope from Step 1 and one of our points to plug into the point-slope formula: \[y - y_1 = m(x - x_1)\] If we choose to use the point (3,1), we substitute \(x_1\)=3, \(y_1\)=1, and \(m\)=-3 into the formula to get: \[y - 1 = -3(x - 3)\] which simplifies to: \[y = -3x + 10\]

Step 3: Convert To Standard Form

The last step is to convert our slope-intercept form equation into standard form. The standard form of a linear equation is \[Ax+Bx=C\], where A, B and C are integers, and A>0. When we rearrange our equation to this form, we get: \[3x+y =10\]

Key Concepts

Slope Calculation

Slope is a measure of how steep a line is and is an essential concept in coordinate geometry. It is calculated by finding the difference in the y-coordinates (rise) and dividing it by the difference in the x-coordinates (run) between two points on a line. Mathematically, this is expressed as

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

For the points \((3,1)\) and \((4,-2)\), the slope \(m\) is determined by substituting into the formula: \( m = \frac{-2 - 1}{4 - 3} = -3 \). The negative value indicates that the line is decreasing; as you move from left to right, the line goes down.

Point-Slope Formula

The point-slope formula is a method employed to form an equation of a line when you know the slope and one point on the line. Using the formula

\( y - y_1 = m(x - x_1) \)

where \(m\) represents the slope and \((x_1, y_1)\) is a point on the line, we can construct a linear equation. Substituting the values from the earlier slope calculation and using \((3,1)\) as our point, we arrive at: \( y - 1 = -3(x - 3) \), which simplifies to \( y = -3x + 10 \), giving us the equation in slope-intercept form.

Linear Equations

Linear equations are algebraic expressions that represent straight lines when plotted on a coordinate plane. They typically have one or two variables and the highest degree of any variable is one. There are several forms of linear equations, with slope-intercept form (\(y = mx+b\)) and standard form (\(Ax + By = C\)) being the most common.

To transform the slope-intercept form to standard form, you usually move terms involving the variables to one side of the equation and the constant to the other while ensuring the coefficients are integers, and the coefficient for \(x\) is positive.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, connects algebra and geometry through the use of a coordinate system. It allows us to solve geometrical problems using algebraic equations. For instance, the equation of a line can be determined and graphed using specific points, slopes, and formulas.

Acquainting oneself with the basics such as calculations for slope, different forms of linear equations including point-slope and standard form, is necessary for understanding and solving problems in this branch of mathematics. Furthermore, the graphical representation of linear equations can assist in visualizing the relationship between variables.