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Lewis is saving for a down payment of $[tex]$8,750$[/tex] on a house. The initial amount of money he has saved is $[tex]$250$[/tex], and the amount of money he plans on saving every month is $[tex]$340$[/tex]. To model the situation, he created this equation, where [tex]$x$[/tex] represents the number of months and [tex]$y$[/tex] represents the total amount of savings:
[tex]$y=250+340 x$[/tex]
Determine the missing steps and justifications.
[tex]$\begin{array}{l}
\text { Subtraction property of equality } \quad y-250=340 x \quad \text { division property of equality } \quad \frac{y-250}{340}=x \
\begin{array}{lll}
\frac{8,750-250}{340}=25=x & y-340=250 x & \frac{8,750-340}{250}=34=x \quad \text { multiplication property of equality }
\end{array}
\end{array}$[/tex]
| Steps | Justification |
| ----------- | ----------- |
| [tex]$y=250+340 x$[/tex] | given |
| [tex]$y-250=250+340 x-250$[/tex] | |
| | simplification |
| [tex]$\frac{y-250}{340}=\frac{340 x}{340}$[/tex] | |
| | simplification |
| | substitution and simplification
Answer (2)
Subtract 250 from both sides of the equation: y − 250 = 340 x .
Divide both sides by 340: 340 y − 250 = x .
Substitute y = 8750 : 340 8750 − 250 = x .
Simplify to find the number of months: x = 25 .
Explanation
Problem Analysis We are given the equation y = 250 + 340 x , where y represents the total amount of savings, and x represents the number of months. Lewis wants to save $8 , 750 for a down payment. Our goal is to find the number of months it will take him to reach his goal. We will fill in the missing steps and justifications in the table.
Subtracting 250 from both sides The first step is to subtract 250 from both sides of the equation to isolate the term with x . This gives us y − 250 = 250 + 340 x − 250 . The justification for this step is the subtraction property of equality.
Simplification Simplifying the equation y − 250 = 250 + 340 x − 250 , we get y − 250 = 340 x .
Dividing both sides by 340 Next, we divide both sides of the equation by 340 to solve for x . This gives us 340 y − 250 = 340 340 x . The justification for this step is the division property of equality.
Simplification Simplifying the equation 340 y − 250 = 340 340 x , we get 340 y − 250 = x .
Substitution and Simplification Finally, we substitute y = 8750 into the equation 340 y − 250 = x to find the value of x . This gives us 340 8750 − 250 = x . Simplifying, we have 340 8500 = x . Dividing 8500 by 340, we get x = 25 .
Examples
Understanding how to save for a down payment on a house is a practical application of linear equations. For example, if you want to buy a car that costs $15 , 000 and you have $3 , 000 saved, and you plan to save $500 per month, you can use the equation y = 3000 + 500 x to determine how many months it will take to save enough money. Substituting y = 15000 , we get 15000 = 3000 + 500 x . Solving for x , we find x = 24 months. This shows how linear equations can help you plan and achieve your financial goals.
Lewis can calculate that it will take him 25 months to save $8,750 for a down payment on a house using the equation y = 250 + 340 x . By isolating the variable and substituting the target savings amount, we find that x = 25 . This result is achieved through a series of algebraic steps including subtraction and division properties of equality.
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