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Magazine Chapter 4: Problem 70
Solve the system of equations by elimination and explain all your steps in words: \(\left\\{\begin{array}{l}5 x+4 y=10 \\ 2 x=3 y+27\end{array}\right.\).
Short Answer
Expert verified
x = 6, y = -5
Step by step solution
01
- Rewrite the Second Equation
Rewrite the second equation so that it has a similar format to the first equation. Start with the original second equation: \[2x = 3y + 27\]Rearrange it to match the form of the first equation: \[2x - 3y = 27\]
02
- Align the Equations
Place both equations one below the other for easier elimination: \[5x + 4y = 10\]\[2x - 3y = 27\]
03
- Multiply the Equations to Facilitate Elimination
Find the least common multiple (LCM) of the coefficients of either the x terms or y terms to eventually eliminate one of the variables. We'll choose to eliminate x. The LCM of 5 and 2 is 10, so multiply the entire first equation by 2 and the second equation by 5:\[2(5x + 4y) = 2(10)\] -> \[10x + 8y = 20\]\[5(2x - 3y) = 5(27)\] -> \[10x - 15y = 135\]
04
- Subtract the Equations
Subtract the second modified equation from the first to eliminate x:\[ (10x + 8y) - (10x - 15y) = 20 - 135\]This simplifies to:\[ 23y = -115\]
05
- Solve for y
Solve for y by dividing both sides of the equation by 23:\[ y = \frac{-115}{23} = -5\]
06
- Substitute y into Original Equation
Substitute y = -5 back into one of the original equations to solve for x. Using the first equation:\[5x + 4(-5) = 10\] -> \[5x - 20 = 10\]Add 20 to both sides:\[5x = 30\]Divide by 5:\[x = 6\]
07
- Write the Solution
The solution to the system of equations is:\[ x = 6, y = -5 \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
elimination method
The elimination method is a powerful tool for solving systems of linear equations. The primary goal is to eliminate one of the variables by adding or subtracting the equations from each other. This process makes it easier to solve for one variable and subsequently use its value to find the other variable. Let's break down the process:
1. **Rewrite the Equations**: First, ensure that both equations are in a similar format. This step might involve rearranging terms.
2. **Align the Equations**: Arrange them one on top of the other to visually keep track of your work.
3. **Find the Least Common Multiple (LCM)**: Multiply each equation so that one variable has the same coefficient in both equations, making elimination straightforward.
4. **Eliminate a Variable**: Add or subtract the equations to eliminate one of the variables.
Now, you will have a single-variable equation which can be easily solved. Finally, substitute the value back into one of the original equations to find the remaining variable.
1. **Rewrite the Equations**: First, ensure that both equations are in a similar format. This step might involve rearranging terms.
2. **Align the Equations**: Arrange them one on top of the other to visually keep track of your work.
3. **Find the Least Common Multiple (LCM)**: Multiply each equation so that one variable has the same coefficient in both equations, making elimination straightforward.
4. **Eliminate a Variable**: Add or subtract the equations to eliminate one of the variables.
Now, you will have a single-variable equation which can be easily solved. Finally, substitute the value back into one of the original equations to find the remaining variable.
linear equations
Linear equations form the basis of many algebraic problems. A linear equation is an equation that makes a straight line when graphed on a coordinate plane. It is generally of the form: ax + by = c, where a, b, and c are constants.
Each solution to a linear equation is a combination of x and y that makes the equation true. For example, the equation 5x + 4y = 10 can be graphed as a line where any point (x, y) on the line satisfies the equation.
When working with a system of linear equations, such as:5x + 4y = 102x = 3y + 27,
our goal is to find the point (x, y) that satisfies both equations simultaneously. This means finding where the two lines intersect.
Mastering linear equations is crucial, as they are foundational not just in algebra, but also in higher mathematics and many real-world applications.
Each solution to a linear equation is a combination of x and y that makes the equation true. For example, the equation 5x + 4y = 10 can be graphed as a line where any point (x, y) on the line satisfies the equation.
When working with a system of linear equations, such as:5x + 4y = 102x = 3y + 27,
our goal is to find the point (x, y) that satisfies both equations simultaneously. This means finding where the two lines intersect.
Mastering linear equations is crucial, as they are foundational not just in algebra, but also in higher mathematics and many real-world applications.
substitution
Substitution is another effective method for solving systems of equations. This method involves solving one of the equations for one variable and substituting that value into the other equation. Here's a step-by-step outline:
1. **Isolate a Variable**: Solve one of the equations for one of the variables in terms of the other.
2. **Substitute**: Insert this expression into the other equation, replacing the isolated variable.
3. **Solve for the Second Variable**: You now have an equation with one variable, which you can solve.
4. **Back-Substitute**: Take the solution and substitute it back into the original equation to find the remaining variable.
In our example, after determining y = -5 and substituting that back into one of the original equations, we were able to solve for x = 6. Both methods, elimination and substitution, are reliable and often complementary techniques in solving systems of linear equations.
1. **Isolate a Variable**: Solve one of the equations for one of the variables in terms of the other.
2. **Substitute**: Insert this expression into the other equation, replacing the isolated variable.
3. **Solve for the Second Variable**: You now have an equation with one variable, which you can solve.
4. **Back-Substitute**: Take the solution and substitute it back into the original equation to find the remaining variable.
In our example, after determining y = -5 and substituting that back into one of the original equations, we were able to solve for x = 6. Both methods, elimination and substitution, are reliable and often complementary techniques in solving systems of linear equations.
