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Chapter 7: Problem 21

Solve each system by the substitution method. Be sure to check all proposed solutions. \(\left\\{\begin{array}{l}2 x-y=3 \\ 5 x-2 y=10\end{array}\right.\)

Short Answer

Expert verified
The solution to the system is \(x = 4\) and \(y = 5\).

Step by step solution

01

Isolate a Variable

We start by isolating one of the variables in one of the equations. Here, we will isolate the variable \(y\) in the first equation. This yields: \(y = 2x - 3\)
02

Substitute the isolated variable into the second equation

Now, substitute \(y\) into the second equation. This gives:\(5x - 2(2x - 3) = 10\)Solving this equation gives the value for \(x\). We get: \(5x - 4x + 6 = 10\) this simplifies to: \(x = 10 - 6 = 4\)
03

Substitute \(x\) value into the equation from Step 1

Substitute \(x = 4\) back into \(y = 2x - 3\) to find \(y\):\(y = 2(4) - 3 = 5\)
04

Check the Solution

We check the solution by plugging the obtained \(x\) and \(y\) values into both original equations. Equation 1: \(2*4 - 5 = 3\) and Equation 2: \(5*4 - 2*5 = 10\). In both cases, the equations hold true, confirming the solution to be correct.

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Most popular questions from this chapter

a. Create a scatter plot for the data in each table. b. Use the shape of the scatter plot to determine if the data are best modeled by a linear function, an exponential function, a logarithmic function, or a quadratic function. $$ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 4 \\ \hline 1 & 5 \\ \hline 2 & 7 \\ \hline 3 & 11 \\ \hline 4 & 19 \\ \hline \end{array} $$

Compare the graphs of \(3 x-2 y>6\) and \(3 x-2 y \leq 6\). Discuss similarities and differences between the graphs.

Graph each linear inequality. \(x \leq-4\)

Graph each linear inequality. \(3 x+4 y \leq-12\)

Graph each linear inequality. \(2 x+3 y>12\)
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