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Chapter 7: Problem 4

Solve the simultaneous equations $$ 3 x-2 y=11,5 x+7 y=39 $$

Short Answer

Expert verified
Answer: The values for \(x\) and \(y\) that satisfy the given simultaneous equations are \(x = 5\) and \(y = 2\).

Step by step solution

01

Write down the equations

Write down the simultaneous equations: $$ 3x - 2y = 11 \hspace{1cm} (1) $$ $$ 5x + 7y = 39 \hspace{1cm} (2) $$
02

Solve one equation for one variable

We will solve equation (1) for \(x\): $$ x = \frac{11 + 2y}{3} $$
03

Substitute the result into the other equation

Now substitute the result from Step 2 into equation (2): $$ 5\left(\frac{11 + 2y}{3}\right) + 7y = 39 $$
04

Solve the resulting equation for the remaining variable

Simplify the equation and solve for \(y\): $$ \frac{55 + 10y}{3} + 7y = 39 $$ Multiply both sides by 3 to eliminate the fraction: $$ 55 + 10y + 21y = 117 $$ Combine like terms: $$ 31y = 62 $$ Now, solve for \(y\): $$ y = \frac{62}{31} $$ $$ y = 2 $$
05

Substitute the found value back into the equation from Step 2

Substitute the value of \(y\) we found in Step 4 into the equation for \(x\), found in Step 2: $$ x = \frac{11 + 2(2)}{3} $$ $$ x = \frac{15}{3} $$ $$ x = 5 $$
06

Check the solution in both given equations

Check the values \(x = 5\) and \(y = 2\) in both equation (1) and equation (2). For equation (1), \(3x - 2y = 11\): $$ 3(5) - 2(2) = 11 $$ $$ 15 - 4 = 11 $$ $$ 11 = 11 $$ For equation (2), \(5x + 7y = 39\): $$ 5(5) + 7(2) = 39 $$ $$ 25 + 14 = 39 $$ $$ 39 = 39 $$ Both equations are true, so the solution is \(x = 5\) and \(y = 2\).

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Most popular questions from this chapter

Verify that the given value is a solution of the given equation. $$ 11 x-1=10, x=1 $$

Express in partial fractions \(\frac{x^{3}+x+1}{x^{2}+7 x+12}\).

Solve the inequality \(|3 x+2| \leq 4\)

For each fraction state the degrees of the numerator and denominator, and hence determine which are proper and which are improper: (a) \(\frac{x+1}{x}\) (b) \(\frac{x^{2}}{x^{3}-x}\) (c) \(\frac{(x-1)(x-2)(x-3)}{x-5}\)

In each case verify that the given values satisfy (b) \(x=4, y=3\) satisfy \(x+y=7\) and the given simultaneous equations: (a) \(x=2, y=-2\) satisfy \(7 x+y=12\) and (c) \(x=-3, y=2\) satisfy \(8 x-y=-26\) \(-3 x-y=-4\) and \(9 x+2 y=-23\)
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