In Exercises 49–50, solve each system for x and y, expressing either value in terms of a or b, if necessary. Assume that a ≠ 0, b ≠ 0 5ax + 4y = 17 ax + 7y = 22

Verified step by step guidance

1. Start with the system of equations: $\mathrm{5ax}+\mathrm{4y}=17\mathrm{ax}+\mathrm{7y}=22$.
2. To solve for $x$ and $y$, use the method of elimination or substitution. Here, elimination is convenient. Multiply the second equation by 5 to align the coefficients of $\mathrm{ax}$: $5(ax)\; +\; 5(7y)\; =\; 5(22)which\; simplifies\; to5ax\; +\; 35y\; =\; 110.$
3. Now subtract the first equation from this new equation to eliminate $\mathrm{5ax}$: $(5ax\; +\; 35y)\; -\; (5ax\; +\; 4y)\; =\; 110\; -\; 17which\; simplifies\; to31y\; =\; 93.$
4. Solve for $y$ by dividing both sides by 31: $\mathrm{y\; =\; \backslash (\backslash frac\{93\}\{31\}\backslash )}$. Since the problem asks to express in terms of $a$ or $b$ if necessary, check if any variables remain; here, $y$ is a constant.
5. Substitute the value of $y$ back into one of the original equations, for example, $\mathrm{ax\; +\; 7y\; =\; 22}$, to solve for $x$. Rearrange to isolate $x$: $\mathrm{ax\; =\; 22\; -\; 7y}$, then $\mathrm{x\; =\; \backslash (\backslash frac\{22\; -\; 7y\}\{a\}\backslash )}$. This expresses $x$ in terms of $a$.

Key Concepts

Solving Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. The goal is to find values for the variables that satisfy all equations simultaneously. Common methods include substitution, elimination, and using matrices. In this problem, solving for x and y involves manipulating the equations to isolate one variable and substitute it into the other.

Expressing Variables in Terms of Parameters

When equations contain parameters like a and b, solutions for variables x and y may be expressed in terms of these parameters. This means the solution is not a single number but a formula depending on a and b. It is important to treat parameters as constants during algebraic manipulation, ensuring conditions like a ≠ 0 are considered to avoid division by zero.

Elimination Method for Systems with Parameters

The elimination method involves adding or subtracting equations to eliminate one variable, making it easier to solve for the other. When parameters are involved, coefficients include variables like a, so careful algebraic manipulation is needed. After eliminating one variable, the remaining equation can be solved for the other variable in terms of the parameters.