Problem 37 - Algebra A Combined Function

Question

Write an equation of each line. Write the equation in standard form unless indicated otherwise. See Examples 1 through \(6 .\) Slope \(2 ;\) through (-2,3)

Step-by-Step Solution

Answer

The line equation is \(2x - y = -7\) in standard form.

Step 1: Understanding the Equation of a Line

The general equation for a line in slope-intercept form is given by \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. We are given the slope \(m = 2\) and a point \((-2, 3)\) through which the line passes.

Step 2: Substituting the Known Values

Substitute the slope \(m = 2\) and the point \((-2, 3)\) into the slope-intercept form equation \(y = mx + b\) to find \(b\). This gives the equation \(3 = 2(-2) + b\).

Step 3: Solving for the y-intercept \(b\)

Solve the equation from Step 2: \(3 = -4 + b\). Add \(4\) to both sides to solve for \(b\): \[ b = 3 + 4 = 7 \] So, the y-intercept \(b\) is \(7\).

Step 4: Writing the Equation in Slope-Intercept Form

Plug \(m = 2\) and \(b = 7\) back into the equation \(y = mx + b\), which gives us:\[ y = 2x + 7 \]

Step 5: Converting to Standard Form

To convert the equation from slope-intercept form \(y = 2x + 7\) to standard form \(Ax + By = C\), first move \(2x\) to the left side:\[ -2x + y = 7 \] Multiply the entire equation by \(-1\) to make \(A\) positive:\[ 2x - y = -7 \]This is the standard form of the equation.

Key Concepts

Slope-Intercept Form

When discussing linear equations, one of the most widely used forms is the slope-intercept form. It is structured as \(y = mx + b\), where:

• \(m\) represents the slope of the line, indicating its steepness and direction – positive means ascending, while negative means descending.
• \(b\) signifies the y-intercept, the point where the line crosses the y-axis.

Understanding this form is crucial as it provides a quick way to graph a line using its slope and y-intercept. You start at the y-intercept \(b\) on the graph, and use the slope \(m\) to find additional points by following the rise over run method. This form is particularly beneficial when you are given a slope and one point on the line or if you wish to understand the behavior of the line at a glance.
To transform an equation into this form, ensure that \(y\) is isolated on one side, leaving a clear view of \(m\) and \(b\). This form directly expresses how \(y\) changes with \(x\).

Standard Form Equation

The standard form of a linear equation is expressed as \(Ax + By = C\) where \(A\), \(B\), and \(C\) are integers, and \(A\) should ideally be positive. Unlike the slope-intercept form, the standard form emphasizes the balance of the equation, which makes it suitable for equations involving both variables simultaneously.
This format is often preferred in computational scenarios, like solving systems of linear equations, because every variable is collected on one side, showing equivalence between expressions. To derive the standard form from the slope-intercept form, rearrange the terms so that \(x\) and \(y\) are on one side of the equation, typically in the form of \(Ax + By = C\). Sometimes, it may require multiplication to clear out fractions or to make \(A\) positive. This reformation gives a clear algebraic look and can make solving or graphing easier in different contexts.
The standard form is especially useful when intercepts need to be calculated as specific integer values.

Y-Intercept

The y-intercept of a line is the value of \(y\) where the line crosses the y-axis. In the slope-intercept form \(y = mx + b\), the y-intercept is represented by \(b\). This particular value is key because it provides a straightforward starting point when graphing a linear equation.
Finding the y-intercept can involve plugging given points or solving for \(b\) when the slope and a point on the line are known. As illustrated, with a known slope \(2\) and point \((-2, 3)\), inserting these into the slope-intercept form equation helps derive \(b\) by calculation:

• Start with \(y = mx + b\)
• Substitute the values: \(3 = 2(-2) + b\)
• Solve: \(b = 3 + 4 = 7\)

The y-intercept helps with predictions about the line's behavior, especially in scenarios without a graph available, providing insights into how the line approaches the y-axis.

Solving Equations

Solving equations is a fundamental skill in algebra that involves finding the values of variables that satisfy the equation. For linear equations, the goal is often to express variables in terms of each other, such as converting between different forms like the slope-intercept and standard forms.
For example, when given a slope \(m\) and a single point, solving may involve determining \(b\) in the slope-intercept form by isolating and computing it from the equation. In more complex contexts, solving might include:

• Rearranging terms to simplify equations, such as isolating a variable.
• Balancing both sides of an equation by adding, subtracting, multiplying, or dividing like terms.

After finding expressions or integers for unknown quantities, solutions are often validated by substituting back into the original equation to ensure consistency. This process sharpens problem-solving approaches necessary to maneuver through diverse algebraic challenges.