Problem 135 - Elementary Algebra

Question

In the following exercises, solve the systems of equations by substitution. $$ \left\\{\begin{array}{l} 5 x+2 y=1 \\ -5 x-4 y=-7 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \(x = -1\) and \(y = 3\).

Step 1 - Solve for one variable

Choose one of the equations to solve for one variable. Let's solve the first equation for x.\[5x + 2y = 1\]Subtract 2y from both sides:\[5x = 1 - 2y\]Divide by 5:\[x = \frac{1 - 2y}{5}\]

Step 2 - Substitute into the second equation

Now, substitute \(x = \frac{1 - 2y}{5}\) into the second equation:\[-5x - 4y = -7\]Substitute:\[-5\left( \frac{1 - 2y}{5} \right) - 4y = -7\]Simplify:\[-(1 - 2y) - 4y = -7\]\[-1 + 2y - 4y = -7\]\[-1 - 2y = -7\]

Step 3 - Solve for y

Solve the simplified equation for y:Add 1 to both sides:\[-2y = -6\]Divide by -2:\[y = 3\]

Step 4 - Find x using y

Now that we have y, substitute it back into the equation for x:\[x = \frac{1 - 2y}{5}\]Substitute y = 3:\[x = \frac{1 - 2(3)}{5} \]\[x = \frac{1 - 6}{5}\]\[x = \frac{-5}{5}\]\[x = -1\]

Step 5 - Check the solution

Check your solution by substituting x and y into the original equations:For the first equation:\[5(-1) + 2(3) = -5 + 6 = 1\]For the second equation:\[-5(-1) - 4(3) = 5 - 12 = -7\]Both equations are satisfied.

Key Concepts

Substitution Method

The substitution method is a technique for solving systems of equations. Here's how it works:
Pick one of the equations in the system. Solve this equation for one of the variables.
In our example, we solve for x from the first equation: \[5x + 2y = 1 \]
After isolating x, we get: \[x = \frac{1 - 2y}{5} \]
Substitute this expression for x into the other equation. This will allow us to solve for y.
Substitution method is particularly useful when one equation is easy to solve for one variable.
In our example, we could solve for x in just a few steps! By substituting back, we maintained the solution consistency of the overall system.

Solving Linear Equations

Linear equations represent straight lines. Solving them means finding the values of variables that satisfy the equations.
In our system, we have two linear equations: \[5x + 2y = 1\] \[-5x - 4y = -7\]
By solving these equations, we find the intersection of the two lines. This gives us the solution to the system.
For our example, solving directly for y helped make the process straightforward: \[-1 - 2y = -7 \]
We continued simplifying to find y, and eventually found y = 3.
Simplify your calculations step by step and check each part to ensure accuracy.

Step-by-Step Algebra

Approaching algebra step-by-step helps to ensure understanding and accuracy.
We start by solving one variable, as seen here: \[5x = 1 - 2y \] Next, substitute this into the other equation. Simplifying this, we found: \[-1 - 2y = -7\]
Divide to isolate y, giving y = 3. Substitute y back to find x: \[x = \frac{1 - 6}{5} = -1\]
Finally, check both solutions in the original equations to confirm they satisfy both conditions.
Breaking it down into steps ensures every part is clear, making algebra manageable and even fun!