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Chapter 4: Problem 17

(a) solve by substitution. (b) if there is one solution, check. $$ \begin{aligned} &5 x-2 y=-6 \\ &y=x \end{aligned} $$

Short Answer

Expert verified
The solution is \( x = -2 \) and \( y = -2 \). The solution is verified.

Step by step solution

01

Substitute the value of y

Given the second equation, substitute the value of y into the first equation. Since \( y = x \), replace y in the first equation with x.
02

Form a single-variable equation

The first equation is \( 5x - 2y = -6 \). After substituting \( y = x \), the equation becomes \( 5x - 2x = -6 \).
03

Simplify the equation

Combine like terms in the equation \( 5x - 2x = -6 \) to get \( 3x = -6 \).
04

Solve for x

Divide both sides of the equation \( 3x = -6 \) by 3 to isolate x. This gives \( x = -2 \).
05

Find the value of y

Since \( y = x \), substitute \( x = -2 \) back into the equation \( y = x \). This gives \( y = -2 \).
06

Check the solution

Substitute \( x = -2 \) and \( y = -2 \) back into the original equations to verify the solution. The first equation is: \( 5(-2) - 2(-2) = -10 + 4 = -6 \), which is true. The second equation is: \( y = x \), which gives \( -2 = -2 \), also true.

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Key Concepts

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

Solving Linear Equations
Linear equations are mathematical expressions that represent a straight line when graphed. In these problems, you are solving for the unknown variable that makes the equation true. For example, the equation 5x - 2y = -6 can be worked out step by step to find the values of x and y.
  • First, substitute values using the given relationship (like y = x).
  • Next, simplify the equation to combine like terms.
  • Finally, solve for the single variable by isolating it using algebraic operations such as addition, subtraction, multiplication, or division.
This step-by-step process makes it easier to handle more complex equations by breaking them down into simpler tasks.
Single-Variable Equations
Single-variable equations involve just one unknown variable, making them simpler to solve than multi-variable equations. When solving these equations, follow these steps:
  • First, isolate the variable by performing the same operation on both sides of the equation.
  • Depending on the equation, this might mean adding, subtracting, multiplying, or dividing both sides.
For instance, in the equation 3x = -6, you can find x by dividing both sides by 3, leading to x = -2.
Always ensure that you perform each operation correctly to maintain the equation's balance and get the accurate value of the variable.
Checking Solutions
Once the values of the unknown variables are found, it is essential to check the solutions to confirm their correctness.
  • Substitute the found values back into the original equations.
  • Perform the necessary calculations to verify if both sides of the equations are equal.
In this example, substituting x = -2 and y = -2 back into the original equations:
  • The first equation: 5(-2) - 2(-2) = -10 + 4 = -6 holds true, confirming the solution.
  • The second equation: y = x, leading to -2 = -2, is also correct.
Verification is crucial because it eliminates any doubt about the solution's accuracy and ensures a solid grasp of the procedure.

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Most popular questions from this chapter

The maximum combined length and girth of a package that is mailed at the priority rate with the U.S. Postal Service is \(108 \mathrm{in}\). The length is the measure of the longest side of the package, and the girth is the distance measured around the thickest part of the parcel. A box is 3 in. high, 15 in. long, and 8 in. wide. Find its girth.

For exercises 29-34, a karat describes the percent gold in an alloy (a mixture of metals). $$ \begin{array}{|c|c|} \hline \text { Name of alloy } & \text { Percent gold } \\ \hline \text { 10-karat gold } & 41.7 \% \\ \text { 14-karat gold } & 58.3 \% \\ \text { 18-karat gold } & 75 \% \\ \text { 20-karat gold } & 83.3 \% \\ \text { 24-karat gold } & 100 \% \\ \hline \end{array} $$ Find the amount of 14-karat gold and the amount of 20 -karat gold to combine to make 8 oz of \(18-k a r a t\) gold. Round to the nearest hundredth.

For exercises \(45-48\), determine whether the ordered pair is a solution of the system. $$ \begin{gathered} (2,4) ; 2 x+y \leq 30 \\ y \geq 2 x \\ x \geq 1 \end{gathered} $$

For exercises 41-76, use the five steps and a system of equations. The cost of two cans of chili and three cans of soup is $$\$ 7.78$$. The cost of five cans of chili and four cans of soup is $$\$ 12.45$$. Find the cost of one can of chili. Find the cost of one can of soup.

Mixture A is \(15 \%\) sugar. Mixture B is \(18 \%\) sugar. Find the amounts of Mixture A and water needed to make \(20 \mathrm{~L}\) of a new mixture that is \(12 \%\) sugar.
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