Problem 79 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write a linear equation that has the given solution. (There are many correct answers.) $$x=-3$$

Step-by-Step Solution

Answer

A linear equation for the given solution \(x = -3\) could be \(y = x\).

Step 1: Understanding of Linear Equations

A linear equation is represented by the formula \(y = mx + c\), where \(m\) is the slope of the line (ratio of rise over run), \(c\) is the y-intercept (the point where the line crosses the y-axis, or the value of \(y\) when \(x\) is 0), and \(x\) and \(y\) are the variables. For this task, any values can be selected for \(m\) and \(c\) as long as the equation holds true when \(x = -3\). As there can be many correct answers, let's select \(m = 1\) and \(c = 0\) for simplicity.

Step 2: Substituting the Values and Formulating the Equation

Substituting the chosen values of \(m\) and \(c\) into the equation, we get \(y=1*(-3) + 0\) resulting in \(y = -3\). Hence, the equation \(y = x\) is a valid linear equation with the solution \(x = -3\).

Step 3: Verification of the Solution

To verify, if we substitute \(x = -3\) in our final equation \(y = x\), we are left with \(y = -3\), which is indeed correct, verifying that our linear equation is correct.

Key Concepts

Slope-Intercept Form

A linear equation is often expressed in the slope-intercept form, which is written as \(y = mx + c\). Here, \(m\) represents the slope, and \(c\) represents the y-intercept. The slope \(m\) dictates how steep the line is. It's the ratio of the vertical rise to the horizontal run.

• When \(m > 0\), the line slopes upwards.
• When \(m < 0\), the line slopes downwards.
• When \(m = 0\), the line is horizontal.

With the chosen values \(m = 1\) and \(c = 0\), the equation becomes \(y = x\). Here, the slope is 1, meaning for every unit increase in \(x\), \(y\) increases by the same amount. This results in a line that equally rises along the \(y\) axis as much as it runs along the \(x\) axis.

Solution Verification

Solution verification is a crucial step in confirming that the equation accurately represents the original conditions. After determining the linear equation, substitute the given value of \(x\) back into it to see if it satisfies the equation.
For instance, in the exercise, \(x = -3\) was substituted into \(y = x\), resulting in \(y = -3\).

• The process ensures the line correctly aligns with the specified solution.
• It checks our calculations were performed correctly.

Whenever you're unsure, revisiting the equation by plugging values back in can reassure that the work done is precise and meets the requirements.

Y-Intercept

The y-intercept, represented by \(c\) in the slope-intercept form \(y = mx + c\), is the point where the line crosses the \(y\)-axis.

• Conceptually, it's the value of \(y\) when \(x = 0\).
• It can often be used to easily sketch graphs of equations.

In the equation \(y = x\), the y-intercept is \(0\), meaning the line passes through the origin (0,0). This characteristic is crucial in identifying how the line is positioned on a graph.For more complex equations where \(c\) isn't zero, the y-intercept helps in predicting the initial height of the line on the graph.