Math Problem Statement
Solution
To determineTo the solve the solution to system the of equations given system:
of$$equations:
y\ =[ -yx + = - 3x + \ quad \3text \quad{and} \text{ \andquad} y \ = 4quad yx = - 4x2 - \2]
, we$$
willwe determine where solve the for two the equations point intersect \ by(( solving forx , yx)\) where and the ( twoy lines\ intersect).
.
Step### 1: Step Set the two1 equations: Set equal the two equations equal to to each each other other
SinceSince both expressions both equations are are equal equal to to \( yy\,), we can we set equate them them equal:
:
-x-x + + 33 = = 44xx - - 22. \$$ ] --- --- ###### Step Step 22:: Solve Solve for $$for \( xx\$$ ) 11.. Add Add $$\( xx \$$) to to both both sides sides to eliminate: $$\3 =(- x\) on the left: \[ 3 = 5x - 52x - 2.$$ 2. Add 2 to both sides : \\] [ 25. = Add \(52x\. )\ to] both3 sides.: Divide\[ by5 =5: $$5xx =\ ] 13. .\ Divide] by \(--- ###5 Step \): 3\:[ Solvex for = \( y1 \) \Sub] stitute--- $$### x Step = 3:1 Solve$$ for into \( eithery equation\ (use) $$Sub ystitute \( =x - =x + 3$$): \[ y = -(1 )1\ +) into either of the original equations. Using \(3 = 2.$$ --- ### Step 4: Verify they solution = -x + 3\): $$y = -(1) + 3 Substitute $$x = 1$$ into the second equation$$ $$y = 2$$ Thus, $$the solution y = 4x - 2$$: $$y = 4(1) is: \[ (1 -, 2)$$ --- ### Step 4: Verify the2 solution = Check2 in. the\ second equation \(y] =Both equations give $$y4 = x - 2$$: \2[ y = \4(1)), - so the2 solution = is correct4. ---- ### Final Answer: The solution is **$$(1, 2)$$**, which is2: = 2 \] Both equations are** satisfiedB,. so ( the1 solution, is correct . 2--- )**###. Final--- Let Answer me: knowThe if you need further solution details to! the system is **(1, Here are some related questions: 2)**1., How which corresponds do you to ** graphoption B systems of**. linear--- Would equations you like me to to find their intersection? 2 go. over any What methods steps in can more detail? Let me know if you need clarification! Here are 5 related questions to practice: 1. Solve the system $$y = 2x + 1$$ and \(y = -x + 4\ be). 2. Solve \(3 used to solvex systems of equations (graphing, substitution -, elimination)? 3. What is y the importance of = verifying solutions in a system of7 equations? 4\. How do you) check if and two lines are parallel, intersecting, or coincident? 5. How would the solution change if the equations were modified slightly? ** \(Tipx:** Always + 2 check your solution byy = substit uting5 back into\ *both).* original equations!3. What does it mean for two lines to have no solution? 4. Solve a system of equations graphically. 5. Solve the system $$y = 5x - 3$$ and $$y = -2x + 4$$. **Tip:** Always verify your solution by substituting $$x$$ and $$y$$ into both original equations!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Equations
Systems of Equations
Solving for Intersection
Formulas
Equating two linear equations: -x + 3 = 4x - 2
Theorems
Intersection of two lines occurs where their equations are equal
Suitable Grade Level
Grade 8-10
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