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Chapter 3: Problem 64

Graph equation. Solve for \(y\) first, when necessary. \(14 x+7 y=28\)

Short Answer

Expert verified
The equation solved for \(y\) is \(y = 4 - 2x\).

Step by step solution

01

Isolate the y-term

We start with the equation \(14x + 7y = 28\). The goal is to solve for \(y\), so we must first isolate the \(7y\) term. We do this by subtracting \(14x\) from both sides of the equation: \(14x + 7y - 14x = 28 - 14x\) This simplifies to: \(7y = 28 - 14x\).
02

Solve for y

Now that the \(7y\) term is isolated, divide each term by 7 to solve for \(y\): \(\frac{7y}{7} = \frac{28}{7} - \frac{14x}{7}\) This simplifies to: \(y = 4 - 2x\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing
Graphing is a powerful visual tool in mathematics. It helps us understand how equations represent relationships between variables. In this exercise, we are looking at graphing the linear equation given by the formula: 14x + 7y = 28. This equation is in standard form, which can be rewritten by solving for one variable (y in this case). The graph of this linear equation will always be a straight line. Here鈥檚 why graphing is important:
  • It provides a visual representation of equations, making understanding relationships easier.
  • By graphing, you can quickly identify the intercepts and slope of the line.
  • It helps in solving problems by looking at how lines and curves interact.
Once we convert the equation to one where y is isolated, we can easily identify these features and plot the graph accordingly. This makes the process of analyzing and understanding equations more intuitive.
Isolating Variables
Isolating variables is a crucial step in transforming equations, especially when graphing or solving them. In our given equation, 14x + 7y = 28, we need to isolate the y-term to solve for y. This process involves steps that simplify the equation to make y the subject:
  • First, remove any terms not involving y. In this case, subtract 14x from both sides.
  • You will have 7y = 28 - 14x after this step.
  • This separation of terms helps focus on one variable, a common useful method in algebra.
Isolating variables makes it easier to input into a function or graph a line. It transforms complex equations into more manageable ones, providing clarity and insights about their behavior.
Solving for y
In the context of the exercise, solving for y means expressing the equation 14x + 7y = 28 in a form where y is by itself. This is often helpful for finding specific solutions or graphing. The steps involved are systematic and simple:
  • After isolating the y-term by getting 7y = 28 - 14x, the next step is division. Divide every term by 7 to simplify.
  • This results in the equation y = 4 - 2x.
  • Now, we have a linear equation in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.
Solving for y like this allows you to quickly identify how changes in x impact y. It's also essential for graphing on a coordinate plane, providing a clear picture of the line's direction and position. The slope (m) indicates how steep the line is, and the y-intercept (b) shows where the line crosses the y-axis.

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Most popular questions from this chapter

For each pair of equations, determine whether their graphs are parallel, perpendicular, or neither. See Example 6 $$ \begin{aligned} &y=\frac{3}{4} x+1\\\ &4 x-3 y=15 \end{aligned} $$

Find an equation of the line that passes through \((2,5)\) and is parallel to the line \(y=4 x-7 .\) Write the equation in slope-intercept form.

Determine whether the lines through each pair of points are parallel, perpendicular, or neither. See Example 8. \((2,4)\) and \((-1,-1)\) \((8,0)\) and \((11,5)\)

For each pair of equations, determine whether their graphs are parallel, perpendicular, or neither. See Example 6 $$ \begin{aligned} &y=-9 x-3\\\ &y=-9 x \end{aligned} $$

Find the slope of a line perpendicular to the line passing through the given two points. See Example \(9 .\) \(\left(\frac{1}{3},-1\right)\) and \(\left(\frac{4}{3},-2\right)\)
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