Write a system of equations for each problem, and then solve...

Write a system of equations for each problem, and then solve the system. Jonathan, a second grader, counted the money in his piggy bank. He had only quarters and dimes. When he added up his money, he had 39 coins worth a total of $\$ 7.50 .$ How many coins of each kind did he have? $$\begin{array}{|c|c|c|}\hline \text { Number } & {\text { Value }} & {\text { Total }} \\ {\text { of Coins }} & {\text { per Coin }} & {\text { Value }} \\ \hline x & {50.25} & {} \\ {y} & {50.10} & {} \\ {\text { 39 }} & {\times \times \times \times} & {57.50} \\ \hline\end{array}$$

Write a system of equations for each problem, and then solve the system.
Jonathan, a second grader, counted the money in his piggy bank. He had only quarters and dimes. When he added up his money, he had 39 coins worth a total of $\$ 7.50 .$ How many coins of each kind did he have?

$$\begin{array}{|c|c|c|}\hline \text { Number } & {\text { Value }} & {\text { Total }} \\ {\text { of Coins }} & {\text { per Coin }} & {\text { Value }} \\ \hline x & {50.25} & {} \\ {y} & {50.10} & {} \\ {\text { 39 }} & {\times \times \times \times} & {57.50} \\ \hline\end{array}$$

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Key Concepts

Systems of Linear Equations

This concept involves setting up two or more linear equations that share common variables, where each equation represents a different constraint. Solving these systems simultaneously allows you to find the values of the unknowns that satisfy all given conditions. This is a foundational skill in algebra used to model and solve real-world problems.

Variable Assignment and Equation Formulation

This concept is about defining variables to represent unknown quantities and then creating algebraic equations based on the relationships and information provided in a problem. By translating verbal statements or practical situations into equations, you form the groundwork for analyzing and solving the problem mathematically.

Solving Methods for Linear Systems

Once a system of linear equations is formulated, various techniques such as substitution, elimination, or matrix methods can be applied to find the solution. The substitution method involves solving one equation for one variable and then substituting that expression into another equation, while the elimination method involves adding or subtracting equations to cancel out one of the variables.

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