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Chapter 3: Problem 62

(a) complete the table of solutions. (b) graph the equation. $$ \begin{aligned} &x-4 y=10\\\ &\begin{array}{|l|l|} \hline x & y \\ \hline 0 & \\ \hline & 0 \\ \hline 6 & \\ \hline \end{array} \end{aligned} $$

Short Answer

Expert verified
Complete the table with points (0, -2.5), (10, 0), and (6, -1). Plot these points and draw the line.

Step by step solution

01

- Solve for y when x = 0

Start with the equation: \( x - 4y = 10 \). Substitute \( x = 0 \) and solve for \( y \):\( 0 - 4y = 10 \).Divide both sides by -4: \( y = -\frac{10}{4} = -2.5 \).
02

- Solve for x when y = 0

Start with the equation: \( x - 4y = 10 \). Substitute \( y = 0 \) and solve for \( x \):\( x - 4(0) = 10 \).\( x = 10 \).
03

- Solve for y when x = 6

Start with the equation: \( x - 4y = 10 \). Substitute \( x = 6 \) and solve for \( y \):\( 6 - 4y = 10 \).Subtract 6 from both sides: \( -4y = 4 \).Divide both sides by -4: \( y = -1 \).
04

- Complete the table

Using the solutions from Steps 1, 2, and 3, complete the table:\(\begin{array}{|l|l|}\hlinex & y \hline0 & -2.5 \hline10 & 0 \hline6 & -1 \hline\end{array}\)
05

- Plot the points

Plot the points (0, -2.5), (10, 0), and (6, -1) on a graph.
06

- Draw the line

Draw a straight line through the plotted points to graph the equation \( x - 4y = 10 \).

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Key Concepts

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

Algebraic Substitution
Algebraic substitution is a powerful technique to solve equations. It simply means plugging in given values for variables into an equation and solving for the unknowns. This allows us to find specific solutions without doing everything at once. In the provided exercise, we used substitution to find different values of y given x, and vice versa. Here's a quick reminder:
  • Start with the given equation: in this case, it's \( x - 4y = 10 \).
  • Substitute the known value into the equation. For example, if you know \( x = 0 \), replace x in the equation.
  • Solve for the unknown variable by isolating it on one side of the equation.
This method is particularly useful when preparing points for graphing or completing tables of values. It makes solving more manageable step by step.
Graphing Linear Equations
Graphing linear equations involves plotting points on a coordinate plane and drawing a line through these points. In the exercise, the equation given is \( x - 4y = 10 \). Here's the step-by-step process:
  • First, complete the table of solutions by finding at least two points that satisfy the equation.
  • Next, plot these points on a graph. Each point is a solution to the equation represented as (x, y).
  • Finally, draw a straight line through these points. For our exercise, the points (0, -2.5), (10, 0), and (6, -1) were plotted. This line represents the set of all solutions to the equation \( x - 4y = 10 \).
Remember, a linear equation will always graph as a straight line. If your points don't line up, double-check your calculations!
Completing Tables of Values
Completing tables of values helps you understand how variables in an equation relate to each other. It involves finding pairs of x and y that satisfy an equation. For linear equations like \( x - 4y = 10 \), follow these steps:
  • Choose a value for one variable (either x or y).
  • Use algebraic substitution to find the corresponding value for the other variable.
  • Record these values in your table.
In our exercise, we used x = 0, y = 0, and x = 6 to find (0, -2.5), (10, 0), and (6, -1). This is a great method for visualizing solutions before graphing, making it easier to confirm your results are correct. Once your table is complete, these points can be plotted to graph the equation.

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Most popular questions from this chapter

For exercises 37-66, use the slope formula to find the slope of the line that passes through the points. \((1,9) ;(3,14)\)

(a) graph the given points, and draw a line through the points. (b) use the graph to find the slope of the line. (c) use the slope formula to find the slope of the line. \((-20,-50) ;(10,30)\)

(a) graph the given points, and draw a line through the points. (b) use the graph to find the slope of the line. (c) use the slope formula to find the slope of the line. \((-30,-40) ;(10,30)\)

Use the slope formula to find the slope of the line that passes through the points. \(\left(\frac{1}{3}, \frac{4}{5}\right) ;\left(\frac{5}{3}, \frac{2}{5}\right)\)

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