Question

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

Short answer

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

Step-by-step solution

Write down the equations

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

Solve one equation for one variable

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

Substitute the result into the other equation

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

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 \(y\): $$ y = \frac{62}{31} $$ $$ y = 2 $$

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: $$ x = \frac{11 + 2(2)}{3} $$ $$ x = \frac{15}{3} $$ $$ x = 5 $$

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 \(y = 2\).

Key concepts

Linear Equations

Linear equations are mathematical expressions where each term is either a constant or the product of a constant and a single variable. In simpler terms, they form a straight line when graphed on a coordinate plane. This means they have no exponent higher than one or anything multiplying variables together.
Linear equations generally come in the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\), \(y\) are variables. In our given problem, we have two linear equations:

• \(3x - 2y = 11\)
• \(5x + 7y = 39\)

These are examples of simultaneous equations, where the goal is to find values of the variables that satisfy both equations at the same time. By understanding how these equations behave, one can predict outcomes based on given conditions.

Substitution Method

The substitution method is a technique for solving simultaneous equations. The idea is to solve one equation for one variable and then substitute this expression into the other equation. This reduces the number of variables, simplifying the problem.
In our problem, we first solve the first equation for \(x\):

• Equation (1): \(3x - 2y = 11\)
• Rearranging for \(x\): \(x = \frac{11 + 2y}{3}\)

Next, we substitute this expression for \(x\) obtained from equation (1) into equation (2):

• Equation (2) becomes \(5\left(\frac{11 + 2y}{3}\right) + 7y = 39\)

This substitution effectively converts our system of equations into one that only involves \(y\), making it easier to solve.

Solving Equations

Once you've substituted one equation into another, you're left with a single equation that usually requires simplification. Solving equations involves isolating the variable, using basic arithmetic to achieve the goal.
After substituting, our equation is:

• \(\frac{55 + 10y}{3} + 7y = 39\)
• Multiply through by 3 to clear the fraction: \(55 + 10y + 21y = 117\)
• Combine like terms: \(31y = 62\)
• Finally, solve for \(y\): \(y = 2\)

With \(y\) found, we back-substitute into the expression for \(x\):

• \(x = \frac{11 + 2(2)}{3}\)
• Simplify to find \(x = 5\)

To ensure the solution is correct, substitute \(x = 5\) and \(y = 2\) back into the original equations. Both should hold true, confirming your solution is accurate.