Solve this system of equations using the substitution method.

[tex]\begin{array}{c}
y=x-1 \\
4 x+8=y \\
([?],\end{array}[/tex]

Substitute y = x − 1 into the second equation.
Solve for x : 4 x + 8 = x − 1 ⇒ 3 x = − 9 ⇒ x = − 3 .
Substitute x = − 3 back into y = x − 1 to find y = − 4 .
The solution is ( − 3 , − 4 ) ​ .

Explanation

Understanding the Problem We are given a system of two equations:

Equation 1: y = x − 1 Equation 2: 4 x + 8 = y
Our goal is to solve for x and y using the substitution method.

Substitution Since we already have y isolated in Equation 1, we can substitute the expression for y from Equation 1 into Equation 2. This gives us:

4 x + 8 = x − 1

Isolating x Now, we solve for x . First, subtract x from both sides of the equation:

4 x − x + 8 = x − x − 1
3 x + 8 = − 1

Continuing to Isolate x Next, subtract 8 from both sides:

3 x + 8 − 8 = − 1 − 8
3 x = − 9

Solving for x Finally, divide both sides by 3:

3 3 x ​ = 3 − 9 ​
x = − 3

Solving for y Now that we have the value of x , we can substitute it back into Equation 1 to find the value of y :

y = x − 1
y = − 3 − 1
y = − 4

Final Answer Therefore, the solution to the system of equations is x = − 3 and y = − 4 . We can write this as an ordered pair: ( − 3 , − 4 ) .

Examples
Systems of equations are used in many real-world applications, such as determining the break-even point for a business. For example, if a company's cost function is y = 4 x + 8 (where x is the number of units produced and y is the total cost) and the revenue function is y = x − 1 (where x is the number of units sold and y is the total revenue), solving this system of equations will give the number of units that need to be produced and sold for the company to break even. In this case, the company needs to produce -3 units and the revenue will be -4. Note that this is a simplified example and in reality, the number of units cannot be negative.

Answered by GinnyAnswer | 2025-07-03

The solution to the system of equations y = x − 1 and 4 x + 8 = y is ( − 3 , − 4 ) . We found this by substituting the value of y from the first equation into the second equation, solving for x , and then finding y . The intersection point of these equations is at ( − 3 , − 4 ) .
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Answered by Anonymous | 2025-07-04