Verified Answer :
- Solve for \(k\) in \(2t + k = 6\) to get \(k = 6 - 2t\).
- Solve for \(x\) in \(4x + 3y = 12\) to get \(x = 3 - \frac{3}{4}y\).
- Solve for \(y\) in \(4x + 3y = 12\) to get \(y = 4 - \frac{4}{3}x\).
- Graph \(4x - 6y = 24\) by finding the intercepts \((0, -4)\) and \((6, 0)\) or using the slope-intercept form \(y = \frac{2}{3}x - 4\).
Explanation
- Problem Analysis
We are tasked with solving three linear equations for specific variables and describing the procedure for graphing a linear equation. Solving for a variable in a linear equation involves using inverse operations to isolate that variable on one side of the equation. Graphing linear equations typically involves identifying key points such as intercepts or using the slope-intercept form to visualize the relationship between variables.
- Solving for k in 2t + k = 6
To solve for \(k\) in the equation \(2t + k = 6\), we subtract \(2t\) from both sides of the equation:
\[2t + k - 2t = 6 - 2t\]
\[k = 6 - 2t\]
Thus, the expression for \(k\) is \(k = 6 - 2t\).
- Solving for x in 4x + 3y = 12
To solve for \(x\) in the equation \(4x + 3y = 12\), we first subtract \(3y\) from both sides to isolate the term containing \(x\):
\[4x = 12 - 3y\]
Next, we divide both sides by 4 to solve for \(x\):
\[x = \frac{12 - 3y}{4}\]
\[x = 3 - \frac{3}{4}y\]
Thus, the expression for \(x\) is \(x = 3 - \frac{3}{4}y\).
- Solving for y in 4x + 3y = 12
To solve for \(y\) in the equation \(4x + 3y = 12\), we first subtract \(4x\) from both sides to isolate the term containing \(y\):
\[3y = 12 - 4x\]
Next, we divide both sides by 3 to solve for \(y\):
\[y = \frac{12 - 4x}{3}\]
\[y = 4 - \frac{4}{3}x\]
Thus, the expression for \(y\) is \(y = 4 - \frac{4}{3}x\).
- Graphing 4x - 6y = 24
To graph the equation \(4x - 6y = 24\), it is helpful to rewrite it in slope-intercept form, \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
Starting with \(4x - 6y = 24\):
Subtract \(4x\) from both sides:
\[-6y = -4x + 24\]
Divide by -6:
\[y = \frac{-4}{-6}x + \frac{24}{-6}\]
\[y = \frac{2}{3}x - 4\]
The slope is \(\frac{2}{3}\) and the y-intercept is \(-4\). Alternatively, we can find the intercepts:
- If \(x = 0\), \(y = -4\), so the y-intercept is \((0, -4)\).
- If \(y = 0\), \(4x = 24\), so \(x = 6\), making the x-intercept \((6, 0)\).
Plotting these two points and drawing a straight line through them allows us to visualize the equation.
Examples
In economics, linear equations are often used to model supply and demand curves. For example, if the demand for a product is given by a linear equation, a manager can solve for the price (the variable) to determine how many units will be sold at different levels. Mastering these algebraic manipulations allows for efficient decision-making when analyzing market trends or budgeting.