(4x+3y=15),(y=x+5) Reviews the graphs of a system of two linear equations in two variables: 4x+3y=15 and y=x+5. Find the solution to both equations.
To find the solution to both equations, we can use the substitution or elimination method.
1. Substitution method:
- Start with the second equation, y = x + 5.
- Substitute this expression for y in the first equation: 4x + 3(x + 5) = 15.
- Simplify: 4x + 3x + 15 = 15.
- Combine like terms: 7x + 15 = 15.
- Subtract 15 from both sides: 7x = 0.
- Divide both sides by 7: x = 0.
- Substitute this value of x back into the second equation: y = 0 + 5 = 5.
- The solution to both equations is x = 0 and y = 5, or (0, 5).
2. Elimination method:
- Multiply both sides of the second equation by 3 to make the coefficients of y in both equations the same: 3y = 3(x + 5) = 3x + 15.
- Rewrite the first equation: 4x + 3y = 15.
- Subtract the second equation from the first equation: 4x + 3y - (3x + 15) = 0.
- Simplify: 4x + 3y - 3x - 15 = 0.
- Combine like terms: x + 3y - 15 = 0.
- Rearrange the equation: x = 15 - 3y.
- Substitute this expression for x back into the second equation: y = (15 - 3y) + 5.
- Simplify: y = 20 - 3y.
- Add 3y to both sides: 4y = 20.
- Divide both sides by 4: y = 5.
- Substitute this value of y back into the second equation: x = 5 + 5 = 10.
- The solution to both equations is x = 10 and y = 5, or (10, 5).
Both methods yield the same solution: (0, 5) or (10, 5).