Systems by Substitution $\newline$ Clark spent a total of $\$ 46$ on a pair of shoes and a new jacket. The shoes cost $\$ 8$ more than the jacket. Write a system of equations to represent the situation. Then find the cost of each irem.

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Q. Systems by Substitution

\newline

Clark spent a total of

\$ 46

on a pair of shoes and a new jacket. The shoes cost

\$ 8

more than the jacket. Write a system of equations to represent the situation. Then find the cost of each irem.

Identify Variables: Identify the variables for the cost of the shoes and the jacket. $\newline$ Let's denote the cost of the jacket as 'j ext{ dollars} ext{.} ext{ }According ext{ }to ext{ }the ext{ }problem, ext{ }the ext{ }shoes ext{ }cost ext{ } ext{\$}\(8 ext{ }more ext{ }than ext{ }the ext{ }jacket, ext{ }so ext{ }the ext{ }cost ext{ }of ext{ }the ext{ }shoes ext{ }will ext{ }be ext{ }' ext{ }j ext{ }+ ext{ } $8$ ext{ }' ext{ }dollars ext{.} ext{ }
Write First Equation: Write the first equation representing the total cost. $\newline$ The total cost of the shoes and the jacket is $\$46$ . So, the first equation is: $\newline$ $j + (j + 8) = 46$
Write Second Equation: Write the second equation representing the relationship between the cost of the shoes and the jacket. $\newline$ The second equation is already given by the problem: the shoes cost $\$8$ more than the jacket. This is represented by the equation: $\newline$ $\text{shoes} = \text{jacket} + 8$ $\newline$ Since we've already defined the cost of the shoes as ' $j + 8$ ', this equation is not needed as a separate equation in the system.
Combine and Solve: Combine the terms in the first equation and solve for $j$ . $j + j + 8 = 46$ $2j + 8 = 46$
Isolate 'j': Subtract $8$ from both sides of the equation to isolate the term with 'j'. $\newline$ $2j + 8 - 8 = 46 - 8$ $\newline$ $2j = 38$
Solve for 'j': Divide both sides of the equation by $2$ to solve for 'j'. $\newline$ $\frac{2j}{2} = \frac{38}{2}$ $\newline$ $j = 19$
Calculate Shoes Cost: Calculate the cost of the shoes using the value of ' extit{j}'. $\newline$ Since the shoes cost $\$$ $8$ more than the jacket, and we've found that the jacket costs $\$$ $19$ , the shoes cost: $\newline$ shoes = $j + 8$ $\newline$ shoes = $19 + 8$ $\newline$ shoes = $27$
Check Solution: Check the solution by substituting the values back into the original equation. $\newline$ $j + (j + 8) = 46$ $\newline$ $19 + (19 + 8) = 46$ $\newline$ $19 + 27 = 46$ $\newline$ $46 = 46$ $\newline$ The solution checks out.