Problem 16 - Understanding Elementary Algebra with Geometry

Question

$$\text { In Exercises } 15-28, \text { solve the system of equations using the substitution method.}$$ $$\left\\{\begin{aligned} x+3 y &=5 \\ y &=4 x-7 \end{aligned}\right.$$

Step-by-Step Solution

Answer

x = 2, y = 1

Step 1: Substitute y

In the second equation, we have an expression for y, which is already isolated: \(y = 4x - 7\). We will substitute this expression for y in the first equation.

Step 2: Substitute into the first equation

Now substitute \(y = 4x - 7\) into the first equation \(x + 3y = 5\). This will give us: \[x + 3(4x - 7) = 5\]

Step 3: Simplify the equation

Distribute the 3: \[x + 12x - 21 = 5\]. Combine like terms: \[13x - 21 = 5\].

Step 4: Solve for x

To isolate x, add 21 to both sides: \[13x - 21 + 21 = 5 + 21\]. Simplify: \[13x = 26\]. Then divide both sides by 13: \[x = 2\].

Step 5: Solve for y using x

We found that \(x = 2\). Substitute this value back into the equation \(y = 4x - 7\) to find y: \[y = 4(2) - 7\]. Simplify: \[y = 8 - 7 = 1\].

Step 6: Verify the solution

To ensure the solution is correct, substitute \(x = 2\) and \(y = 1\) back into the original equations. \. For the first equation: \[2 + 3(1) = 5\]. This simplifies to \[5 = 5\], which is true. For the second equation: \[1 = 4(2) - 7\]. Simplifies to \[1 = 1\], which is true.

Key Concepts

system of equations

A system of equations involves finding values for variables that satisfy more than one equation simultaneously. In our example, we are working with two equations:

• Equation 1: \(x + 3y = 5\)
• Equation 2: \(y = 4x - 7\)

These equations share the same variables, \(x\) and \(y\). Finding a solution means finding an x-value and a y-value that make both equations true at the same time. The point \((x, y)\) where both lines intersect is the solution.

solving linear equations

Solving linear equations means finding the values of variables that make the equation true. For systems of equations, we often solve them simultaneously. When using the substitution method, we solve one of the equations for one variable and then substitute that expression into the other equation. This step-by-step approach helps isolate and solve for each variable.
In our example, we can simplify and solve the system:

• Substitute \(y = 4x - 7\) into the first equation \(x + 3(4x - 7) = 5\)
• Simplify and solve for \(x\) to get \(x = 2\)
• Substitute \(x = 2\) into \(y = 4x - 7\) to solve for \(y\), yielding \(y = 1\)

Thus, the solution to our system of equations is \((2, 1)\).

algebraic substitution

Algebraic substitution is a powerful tool for solving systems of equations. It involves replacing one variable with an equivalent expression from another equation. This method helps to simplify the system, making it easier to find the solution.
Consider our exercise:

• We start with the equation \(y = 4x - 7\), which is already solved for \(y\).
• Next, substitute \(y = 4x - 7\) into the first equation: \(x + 3(4x - 7) = 5\).
• We then simplify and solve for \(x = 2\).
• Finally, substitute \(x = 2\) back to find \(y = 1\).

This straightforward process of substitution makes solving the equations more manageable and leads us to the correct solution. Knowing how and when to use substitution effectively can simplify complex algebra problems immensely.