Problem 2 - Understanding Elementary Algebra with Geometry

Question

Write an equation of the line satisfying the given conditions. Passing through \((2,7)\) with slope \(-3\)

Step-by-Step Solution

Answer

y = -3x + 13

Step 1: Identify the Point-Slope Form

The Point-Slope Form of a linear equation is given by y - y_1 = m(x - x_1), where (x_1, y_1) is a point on the line and m is the slope.

Step 2: Substitute Given Values

Here, the point is (2, 7) and the slope is -3. Substitute these values into the Point-Slope Form equation where x_1 = 2, y_1 = 7, and m = -3: y - 7 = -3(x - 2).

Step 3: Simplify the Equation

Distribute the slope and then simplify: y - 7 = -3x + 6. Add 7 to both sides to isolate y: y = -3x + 6 + 7.

Step 4: Final Equation

Combine the constants on the right-hand side: y = -3x + 13. This is the equation of the line in slope-intercept form.

Key Concepts

Linear Equations

Linear equations are mathematical expressions that describe a flat, straight line on a graph. When written in standard form, they look like this: Ax + By = C. Here, A, B, and C are constants, and x and y are variables. Linear equations are fundamental in algebra and can help us understand relationships between numbers. Key characteristics of linear equations include:

• They graph as straight lines.
• They have constant slopes.
• There are no exponents higher than 1 on the x or y variables.

Understanding linear equations is essential, as they are the foundation for more complex mathematical concepts.

Slope-Intercept Form

The slope-intercept form is a simple way to express linear equations. It is written as y = mx + b. In this form:

• m represents the slope of the line.
• b represents the y-intercept, which is where the line crosses the y-axis.

This form is particularly useful because it clearly shows the slope and the y-intercept, making it easy to graph the equation. For example, from our exercise, we derived the line equation y = -3x + 13. Here, the slope (m) is -3, and the y-intercept (b) is 13. This tells us the line goes down 3 units for every unit it moves to the right, and it crosses the y-axis at y = 13.

Substitution Method

The substitution method is a useful technique for solving systems of equations. It involves solving one equation for one variable and then substituting that solution into another equation. This can simplify the process and help isolate one variable at a time. Here’s how it works:

• Solve one equation for one of the variables.
• Substitute the expression found into the other equation.
• Solve the resulting equation.

In our example, we substituted the known point (2, 7) and the slope -3 into the point-slope form to find the equation of the line. This method is incredibly versatile and can be applied to different types of algebraic problems.

Distribution in Algebra

Distribution in algebra refers to the process of multiplying a single term outside parentheses by each term inside the parentheses. It is a key step in simplifying equations and expressions. The distributive property is expressed as a(b + c) = ab + ac.

• Multiply each term inside the parentheses by the term outside.
• Simplify the resulting expression.

In our exercise, we used the distributive property to simplify the equation y - 7 = -3(x - 2) into y - 7 = -3x + 6. This step is pivotal in turning complex equations into more manageable forms. Distribution ensures that each component of the expression is taken into account.