- The system consists of Equation (1) \(y = -2x + 4\) and Equation (2) \(3y + x = -3\).
- Equation (1) is already in slope-intercept form (\(y = mx + b\)).
- Rearranging Equation (2) by subtracting \(x\) and dividing by \(3\) yields \(y = -\frac{1}{3}x - 1\).
- Statement 1 is true because it correctly identifies the slope-intercept form of the second equation.
- Final Answer: \(\boxed{1}\)
Explanation
- Analyzing the system of equations
The system of equations is given by:
\[ \begin{array}{l} (1) \quad y = -2x + 4 \ (2) \quad 3y + x = -3 \end{array} \]
The slope-intercept form of a linear equation is written as \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept. Let's analyze both equations to see if they fit this form. Equation (1), \(y = -2x + 4\), is already in the slope-intercept form with \(m = -2\) and \(b = 4\).
- Converting the second equation to slope-intercept form
Now we examine Equation (2), which is \(3y + x = -3\). This equation is in standard form. To convert it to slope-intercept form, we need to isolate \(y\).
Start by subtracting \(x\) from both sides of the equation:
\[3y = -x - 3\]
Next, divide every term on both sides by 3 to solve for \(y\):
\[y = \frac{-x}{3} - \frac{3}{3}\]
\[y = -\frac{1}{3}x - 1\]
This shows that the second equation converted to slope-intercept form is indeed \(y = -\frac{1}{3}x - 1\).
- Evaluating the statements
We have two statements to evaluate:
- The second equation converted to slope-intercept form is \(y = -\frac{1}{3}x - 1\).
- Neither equation is in slope-intercept form.
Based on our analysis in step 1, Equation (1) is already in slope-intercept form, so statement 2 is incorrect. Based on our analysis in step 2, statement 1 is correct. Therefore, the true statement is the first one.
Examples
In economics, slope-intercept form is often used to model supply and demand curves. If you have an equation for a demand curve in standard form, converting it to slope-intercept form allows you to easily identify the price intercept and how quantity changes with price (the slope). This simple algebraic rearrangement is a foundational skill for interpreting linear relationships in real-world data analysis and business modeling.